Do numbers exist?
Many quantities of objects and properties exist. But do numbers exist? Three different views on this.
Platonism. The view that numbers are real an are non-spatiotemporal causally inert objects.
Nominalism. The view that numbers are just a way of describing the world.
Fictionalism. The view that numbers don't exist and that all mathematical discourse is false.
Here's a video outlining the three positions and their problems.
Are you a platonist, a nominalst, or a fictionalist?
Platonism. The view that numbers are real an are non-spatiotemporal causally inert objects.
Nominalism. The view that numbers are just a way of describing the world.
Fictionalism. The view that numbers don't exist and that all mathematical discourse is false.
Here's a video outlining the three positions and their problems.
Are you a platonist, a nominalst, or a fictionalist?
Comments (154)
I guess the circumference of a circle was just as many times longer than the diameter before the discovery of irrational numbers as it was after the discovery.
Yes but there is no such thing as a circle in the world. The circle whose circumference divided by its diameter is exactly pi is not any object that can exist in this mortal world of ours. The circle of mathematics is an ideal circle, a pure mental abstraction.
It's the set of points, whatever they are, in the plane, whatever that is, that are all exactly the same distance from some other point.
Before there were humans, there were round-ish things like planets and stars. But there were no circles. There were no circles until human beings came along and conceived them as an abstraction.
Pi already has abstract mathematical existence. But you can't argue that pi is any realer than that by invoking idealized circles. Circles have exactly the same mode of existence as pi: as idealized mathematical abstractions. Circles and pi are contingent on the evolution of abstract reasoning in humans.
tl;dr: Were there circles before humans?
If our space or at least some part of it is continuous and flat then it contains perfect circles. A perfect circle is simply the set of all points in a plane that are the same distance from some point. But even if our space is not continuous and flat, it doesn't mean that a perfect circle is just a mental object. There seems to be no reason why a continuous and flat space couldn't exist; it just would not be the space we live in.
That's a wild assumption with very little evidence for it, and considerable physical evidence against it.
Quoting litewave
Ditto.
But surely you are not making the claim that mathematical Euclidean space is actually the literal truth about the physical world? If not, please explain what you are claiming. And if so ... well, frankly the burden's on you to provide evidence.
Let me give you some thought questions. If the universe the same as Euclidean space, and points in physical space are like points in Euclidean n-space where I don't care what n is, then is the Continuum hypothesis true or false? That is, how many points are in the unit n-cube?
If anyone seriously believed that physical space was Euclidean space, then set theory would become an experimental science. When the physics postdocs start getting grants to determine the truth value of the large cardinal axioms then maybe you'll have a remote hope of my believing this absurd claim.
Is this the line of argument you are putting forward? If not, then what are you saying exactly?
ps -- I apologize if I sound too emphatic. The physical world and mathematical Euclidean space are very different things. The idea of mathematical continuity is an abstraction.
I will also note for the record that even a discrete space can have a notion of continuity. In fact imagine that the universe consists of a lattice of bowling balls with nothing between them. So there is no continuity as you might think of it.
Nevertheless we can put a metric on a discrete space, and say that the distance is 0 from a point to itself, and 1 between any two distinct points. With this metric, every function whose domain is this discrete space is continuous, by the formal definition of continuity. In topological terms this is the discrete topology, in which every subset is an open set. Math gives us the tools to create many universes; but math can't tell us which model is our universe. My guess would be none of them. Or all of them.
But the major point here is that there are no circles in the real world. Do you believe in the physical reality of dimensionless points?
I think that our space is probably quantized, as suggested by contemporary physics. In that case there seem to be no measurable perfect circles in our space because no measurement instrument can penetrate into the minimum length interval (Planck length). Still, we might take the Planck length as further divisible but not "physically", that is, beyond possibility of measurement / physical interaction.
And of course, as I said, there may actually exist continuous Euclidean spaces, even though we don't live in one.
Two oranges are not the same. Numbering the oranges asserts that those oranges are the same and equal to one another. One orange plus another orange is two oranges is not exact but an approximation.
The universe is analogue, numbers are digital. PI is irresolvable because of this contradiction.
There are no straight lines in nature; maths imposes them.
I think the thread is a no brainer.
Do you suppose that the axiom of choice is true in such a space? Then the Banach-Tarski paradox is true as well. Then matter could be created, contrary to the laws of physics.
Is it possible that you (like me, like Kant, like everybody) have a strong intuition of Euclidean space, yet that intuition is simply misleading? And that in fact mathematical Euclidean space is inconsistent with physical reality?
I don't know. If the axiom of choice is consistent with Euclid's axioms then there can be a Euclidean space with axiom of choice. If the negation of the axiom of choice is consistent with Euclid's axioms then there can be a Euclidean space without axiom of choice.
Quoting fishfry
I don't claim that our laws of physics apply in such a space. I don't even claim that such a space is bound up with a time dimension into a spacetime.
Quoting fishfry
Our space is generally not Euclidean but in everyday life the curvature is usually negligible.
Ah. Interesting point, sort of a category mismatch. When you say Euclidean you mean Euclid's axioms of geometry. When I say Euclidean I mean modern Euclidean space [math]\mathbb R^n[/math]. The axiom of choice of course applies to the latter (unless one chooses to accept its negation) but not to the former.
Quoting litewave
It's not the curvature that's the problem, it's the idea of dimensionless points. There's no such thing in physics except as conceptual abstractions.
Dimensionless points are common to both classical and modern definitions of Euclidean space.
How do you justify the idea of dimensionless points as physical entities? Even in an alternate universe?
Is number something realizable in external world that we experience and we try to discuss? 1 apple +1 apple=2 apple. Something just appears in our mind which we can individuate it. It shows a property of the world.
It seems to be no problem in mathematics. What do you mean by "physical"?
The physical world. The object of study of physicists.
Well, according to physicist Max Tegmark, there's no difference between physical and mathematical structures.
That's a speculative idea, not physics. And besides, he's also suggested a stricter idea of the computable universe. If the universe is a computation, then it's not continuous! Because most real numbers are not computable, so the real number line is full of holes. These are interesting ideas but they are not physics.
If you don't understand the distinction between abstract mathematics and the actual, physical world that we live in, that's something you should try to understand. You've fatally weakened your own argument by admitting you don't know the difference between the two.
And do you know the difference? You didn't explain it.
If you regard as "physical" only the world we live in, then I already said that Euclidean space is not the space we live in.
Does that mean numbers are real? May be. I'm not completely sure but there's something odd about how the laws of nature are mathematical. It can't be all coincidence me thinks. Godel's incompleteness theorems, from what I understand, pokes a hole in formal mathematics but I think completeness of math (I don't know how mathematicians call it) is not necessary to understand our world. I could be wrong.
Quoting TheMadFool
I would really like to hear more on what you mean by numbers being passive and active. I'm currently struggling to get my thoughts together on this, but I think we might have very similar views.
Maybe real but not in the same sense as 'that chair is real', rather 'real' as in reflecting the reality of our shared conclusions, as in 'demonstrably real'.
So circles and numbers are the idealised limit of physical reality? They represent perfect symmetry and to "physically exists" means always to be individuated - a "materially" broken symmetry. Therefore mathematical forms are not real. There is only imperfect matter and its approximations of these forms - always inevitably marred by "accidents". Every physical circle is a bit bent. Any collection of things may be given a number, but no two things are actually alike.
This is certainly a familiar ontological view. But it should be troubling that physicists are having such a hard time finding the "real matter" that is limited by these "unreal mathematical forms". Talk of this "mortal world of ours" is to accept a fundamental materiality to being which is proving only to be another idealisation.
To make your position secure, you need "matter" to be something that physicists can actually put their hands upon and show to be real. As it stands, that is not the case. Instead - as argued by ontic structural realism, for instance - the formal aspect of nature seems the more real when it comes to the question of why fundamental particles exist.
Materialism is in metaphysical crisis. So the old Aristotelian story on substance - the one that folk trot out to oppose Platonism - no longer works.
The story is better flipped on its head. Limits are what produce individuated materiality. And without limits, you would just have "a world of pure accidents". A vagueness that is no particular kind of thing at all.
So good old solid matter - when stripped of bounding form - becomes just a realm of "perfect fluctuation". Instead of being individuated and having efficient cause, it becomes a state of completely inefficient cause. :)
Anyway, the point is that if mathematicians don't believe form to be real, well physicists are struggling to find matter to be real. And the best way out of that bind is to look to causality and treat that as the best definition of "physical reality". From there, we can see how limits and accidents make a nice complementary pairing. Limits reduce accidents. But accidents prevent limits being reached.
Reality becomes a pattern produced by the suppression of fluctuations - a constraint on freedoms.
Are numbers real? Well it is certainly true that our models of reality are social constructions. Epistemically, they are only "a useful idea". That is acknowledged in agreeing that we are modelling.
However when it then comes to our ontic commitments as they arise from enquiry into nature, then we begin to appreciate that the materiality and individuation of the world is something we have too readily taken for granted. It just seems perceptually obvious that we exist in a world of solid objects - chockful of their own histories of material accidents. A substance ontology is what we experience, and any mathematical notions about form seem so clearly an abstraction produced by the creative human mind.
But again, physics no longer supports this perceptual belief. It went looking for the real solid stuff that is matter and didn't find it. All it could find was fluctuations bounded by symmetries.
Maybe it is time to believe the physics. :-O
Is there really any question about it?
There are numbers.
They're abstract objects. They're things. Things are what can be referred to.
Things are what facts relate or are about.
"Exist" isn't metaphysically-defined, and so anyone can have their own opinions about what does or doesn't "exist".
Michael Ossipoff
Quoting apokrisis
I said no such thing. Circles and numbers are abstractions. Limits have a technical definition and I would never use that word imprecisely in a mathematical discussion. This is not the first time you've quoted me as saying something I never said.
Quoting apokrisis
I never said that nor do I agree with the statement. Modular forms are said to be the most symmetric mathematical objects but they're beyond me. Circles have lots of symmetries but I don't know what a perfect symmetry would be.
Quoting apokrisis
Not something I would have said nor do I understand what you mean. If someone asked me if I believe that material objects are broken symmetries I'd first try to figure out if the questioner was a crank; and if not, to ask them what they meant by that. Maybe I'd learn something.
Surely I don't have to explain to you the difference between abstract and physical objects. You're just being disingenuous.
Quoting apokrisis
Mathematical forms are real, they're just not physical.
Quoting apokrisis
I would not say that matter approximates forms; rather I'd say that mathematical forms are often (but not always) abstracted from familiar physical objects.
I don't know what you mean by the accident bit.
Quoting apokrisis
In the British usage of the word?
Quoting apokrisis
This I do believe. Unless you go along with Wheeler's idea that there's only one electron that hurries around a lot. I don't think you can distinguish electrons. But of course electrons are right on the border between the physical and the abstract. I do understand your point that saying that physical things are "really there" is a stretch once we get into the higher realms of physics. Still, one can distinguish between a number and a rock, one being abstract and the other physical. Even you would agree to this distinction, yes?
Quoting apokrisis
It should be. It's yours, not mine. But then again the solipsists do seem to have a point.
Surely you can understand that my response was to someone claiming that the number pi proves that numbers are physical or have material existence. I'm not on any soapbox about the ontology of physics. I understand the traps therein.
I really can't comment on the rest of it. If I understand your point (and I so rarely do) it's that if I'm pressed to say what's physical, I'll say a rock. Then you'll ask me about electrons, quarks, strings, and quantum amplitudes, and I'll be forced to admit that I don't really know what a physical thing is. Then you'll say, Aha! Then the number pi is just as real as a rock!
That your point?
Ok. I don't disagree.
But the number pi is a lot different from a rock.
I'm going back into my vat now. It's nice and warm in there.
ps --
Quoting apokrisis
Wait!! It seems you agree with me after all. I completely agree with this statement.
I offered a statement to see how much you might agree with it. The clue was in the question-mark. So when it comes to formal precision, grammatical conventions appear above your paygrade.
Quoting fishfry
So perhaps you can explain the difference. You might discover that it is not as secure as you want to pretend.
Quoting fishfry
Yep. They're mental. Or something.
Oh lordy.
Quoting fishfry
Right. So you accept that when we really get down to brass tacks - fundamental particles - suddenly all this idea vs reality ontology feels insecure. We are right on the border - of a different metaphysics.
But hey, let's get back to the safety of classical atomist ontology. Let's go back to the world as we originally chose to imagine it.
Sounds legit. No one could get confused about things at the level of everyday commonsense, could they?
Oh lordy.
Quoting fishfry
Hmm. But you "prove" that by claiming the reality of material being. And your view of material being is dependent on the fictions of classical physics - the world of substantial objects.
So you are on a soapbox for sure. You are waving the banner for a particular notion of physicalism. And yet you agree also that this particular notion fails when you get down to brass tacks.
A tad "disingenuous", no?
Quoting fishfry
Again, the point is that a ratio like pi and an object like a rock can be treated as if one is a human invention, a mere accidental notion, while the other is indubitably real in being physical and material. But that is just an ontology endorsing a sharply divided dualism.
It is a highly subjective point of view in that you are happy to assign some objects to "the mind", other objects to "the world". And even the slightest questioning of this paradigm sends you into hyperventilating panic. It constitutes a personal assault.
So I am concerned with better approaches to metaphysics. And the proper relation of the forms of mathematics to the materials of physics is central to that inquiry.
Indeed, it has been ever since Ancient Greece.
Quoting fishfry
A little desperate there?
Yes and No!! Yes, numbers are a human invention. But accidental? No. They don't seem that way. Archimedes wasn't hallucinating. There's some mathematical constant pi "out there." This is a deep mystery. Our abstractions are telling us something about the world. We're not sure what.
I don't think you and I disagree all that much.
Quoting apokrisis
Oh I see. I stand accused of being a dualist. And we know how out of favor they are.
My understanding is that we can accommodate abstract mental constructs quite easily within physicalism. Abstractions are thoughts, biochemical processes in my brain.
But thoughts are still different from rocks. Thoughts and rocks are both physical processes, but they have a different character. One doesn't need dualism.
Even thoughts come in two flavors. Thoughts about physical things, and thoughts about abstract things. Thinking about rocks and thinking about pi. Observations of the world versus dreams. Writing history versus writing fiction. Our brains go quite comfortably back and forth between the real and the unreal. Yet sane people alway know the difference.
Great. I respect that you are strong on the mathematics. So I was hoping for a more productive discussion.
Maths is unreasonably effective. It’s abstractions are more than mere intellectual accidents. There must be a reason for their Platonic seeming necessity. So therefore that is why the nature of mathematical truth remains so central to physicalist inquiry.
If we are not sure, we still ought to be exploring with an open mind.
Quoting fishfry
Neuroscience believes thoughts to be informational processes, not biochemical ones. To use the easily abused computational analogy, the "material physics" explains nothing. You could implement the logic of a Turing machine in some system of tin cans and bits of twine.
So a science of the mind definitely does need a dualist physicalism of some kind. There has to be some ontic difference between information and entropy, even if they also arise in some common (mutual) fashion.
But putting that aside, the issue here is the epistemic one of a distinction between observers and observables. Classical physics just presumes that observers are free agents, able to make measurements of reality without disturbing that reality. And this supports the idea that thoughts and rocks are unproblematically separate. Not only are our conceptions of reality a free invention of the human mind, but so do our perceptions of reality enjoy a matching freedom from our ability to invent.
That is, we invent the physics of rock motion. Then rocks have a motion which we can - without getting entangled and changing anything - concretely measure. There is no epistemic concern about the line between what is our ideas and what is reality.
However we now know better. A clean break between observers and observables looks to have become fundamentally impossible.
This epistemic shock doesn't seem to have registered with the mathematical community as far as I can see. The ontological options are still either that maths is a free invention or a perception of Platonic reality. Maths doesn't have to prove itself in the court of the real world, only in the court of logical opinion. It has to conform to the rules of an informational process - the syntax that is grounded in set theory, or category theory, or whatever other fundamental notion of a closed syntactical system happens to be in vogue at the time.
Quoting fishfry
There is nothing so comfortable as a useful habit. Sanity is not having to think, it appears.
But that is simply advising people to give up on physical inquiry. Quantum mechanics is true but seems insane. So don't think about it.
Quoting fishfry
Yes, it is out there as a ratio capturing a primal relation of a physical world with some kind of limit-state perfect symmetry. Let that world be not perfectly flat, let it be non-Euclidean, and the value of pi starts to wander accordingly.
Between the hyperbolic and the hyperspheric, there is only one geometry that is absolutely balanced enough that the value of pi is as stable as far as the eye can see. Whether your circles are big or small, now pi remains always the same.
Whoops. Are we talking about the reality of relations here? How physically abstract. Whoops. Are we talking about the presumed scale-invariance of observables? How mentally abstract.
So pi pops out of reality, out of nature, not by accident but because the very possibility of a "physical relation" has some emergent invariant limit. It arises out of the broken symmetry that is a perfect orthogonality. :)
Thus on the one hand, pi - as a position on the number line - looks the purest accident. Why should it have that exact value? On the other, pi is the identity relation when it comes to a limit notion of orthogonal dimensionality. We might as well just give its value as 1. Everything else that is less perfectly broken can be measured as some difference to that.
Active numbers have causal import. Numbers can cause things to happen. Passive numbers are, very roughly speaking, the effects of active numbers and can be read like a book. My idea of numbers is very rudimentary.
Agreed.
Quoting apokrisis
Of course nothing of the sort is true, but I see where you're going in the next couple of paragraphs so this is not a crucial point.
What's true is this. Computationalism s the claim that the mind (or the universe, in a more grandiose version) is a computation. Now those neuroscientists who are computationalists believe that thoughts are informational processes; and those who aren't, don't.
I hope you will agree with me that this is a true statement about the states of belief of neuroscientists, and that this is NOT a settled issue by any means. If nothing else, if mind is a computation, what's the algorithm? When you bring me some computer code and say, "Here, this is how you implement an mind. It's 875,356 of C++. Some grad student figured it out," then maybe I'll believe you. Till then, the burden of proof is on you.
It's fine if you want to argue from a computationalist point of view. Just so you don't claim that it's the only point of view and that the neuroscientists have all agreed on it. That last part isn't true.
Quoting apokrisis
Yes I understand substrate independence. Any Turing machine computes the same thing whether you implement it with pencil and paper or on a supercomputer or in the wetware of the brain. So if you believe that mind is a computation, then mind is a TM executed in the hardware of the brain. However if you DON'T believe that mind is a computation, you no longer necessarily have substrate independence. I hope you would grant me this. Searle makes the same point. He's said that he thinks mind is a physical but not computational aspect of the brain. I agree with him about that.
Quoting apokrisis
You are describing the distinction in computer science between a program and a computation. Take the Euclidean algorithm to find the greatest common divisor of two integers. By itself, it does nothing. It csn not find the GCD of two integers. The only way to do that is to execute the algorithm on physical hardware. That is a physical process involving an input of energy and an output of heat. Something a physicist could observe and quantify.
So yes we always have that dualism. Where does the algorithm itself live? Well it lived first in Euclid's brain. But isn't Euclid's mind a physical process? His abstract thoughts are physical processes, and his thoughts can be implemented as physical processes. But I don't see why we need dualism.
Quoting apokrisis
I would say that I think this is a little off-target in the sense that it's one complication too many. If we try to weave quantum theory into this we may lose our way. It's hard enough to try to pin down the difference between the abstract and the physical. So I'm not going to try to think about this. You have to start somewhere, and perhaps we could agree that for purposes of this conversation, there is the number pi and there is a rock, and that we don't have to consider their quantum relationship to each other, if any.
Quoting apokrisis
I don't see why it should. When Wiles proved FLT, he didn't say to himself, "Well, quantum theory says that the integers are just like rocks." Why would this come up? Mathematicians do math. Some mathematicians sometimes do philosophy. But when mathematicians do philosophy they're acting as philosophers, not as mathematicians. Wiles doesn't sit around thinking about the nature of the integers. To a number theories, integers are as real as rocks. I doubt Wiles would agree that he's written a work of fiction. Or even give the matter any thought at all.
Quoting apokrisis
Right. Philosophers are rightly concerned with this question. But mathematicians aren't. There is no epistemic shock or even any thoughts at all of philosophy. Some mathematicans care about these things but it's not a requirement of the job.
Quoting apokrisis
Ooh you are on shaky ground here! Gödel told us that math is NOT an informational process! No algorithm can determine the truth of mathematical statements. First you say mind is a computation and then you say that math is. Well we know for a fact that math is NOT a computational process. Perhaps mind isn't either! You know Penrose has made that argument, that incompleteness shows that mind is not a computation. Nobody takes Penrose's argument seriously, but after all he is Sir Roger and the rest of us aren't.
I hope you can see that your computationalist bias may be leading you astray. There are important things in the world that are not computations. Like mathematical truth.
Quoting apokrisis
Well you can drive yourself nuts thinking about this too hard. If every single thing in the world needed to make perfectly logical sense, we couldn't get out of bed in the morning. Life does not make sense! Of course a computationalist like yourself would not understand that. You think we're all just a computer program. That's a silly idea. There I said it. I really disagree with computationalism.
Quoting apokrisis
Jeez that sounds a little mystical. You're saying that Euclidean geometry is the midpoint between elliptic and hyperbolic geometry. Yes this is a true mathematical fact, but it is not mystical. It's just pi. There are plenty of other real numbers out there too. Pi's not that big a deal really. I only used it as a familiar example of a number whose physical existence can be argued against. I didn't mean it in a mystical sense.
Quoting apokrisis
Yes but you're going all woo-woo about a trivial mathematical fact. Well not trivial, non-Euclidean geometry was a big deal when it was discovered. And as Kant noted, we do seem to have an intuition of Euclidean geometry. I'll grant you that. But I think you're making too much of this.
Also I have a picky little pedantic point. The ratio of a circle's circumference to its diameter is different depending on the geometry. But the number pi is always 3.14159... Pi is a particular real number. It's not defined geometrically these days. If we called the circle ratio the foozle, then you can say that in Euclidean geometry the foozle is pi and in non-Euclidean geometry it's not. But pi is always pi. It's a number. It's like asking what's the value of 3 in hyperbolic space. It's 3.
Quoting apokrisis
You and Kant. He was wrong. You're wrong. Euclidean geometry's not special. It's just something we seem to have an intuition of. I'll grant you the psychological and philosophical interest of that fact. But not the mathematical importance.
In short, you are a Euclidean-chauvinist!
Quoting apokrisis
I'll answer that question, if you'll first tell me why 3, a point on the number line by the purist accident, has that exact value. It's a mystery!! Why does 3 have the exact value of 3? It's because it's the number 3. And pi is the number pi. It's just a real number.
It's true that it's the ratio of a circle's circumference to its diameter is pi, but if it were 3 or 47 or 18, you'd be asking why it's that? It's just what it is. The only really interesting thing is that the ratio is always the same no matter what size the circle is! That's the real breakthrough here, that was a great discovery once. [Edit - You made the point that this is only true in Euclidean geometry. Point taken].
Quoting apokrisis
You are really into pi mysticism. What I mean is, what you wrote here is pretty word salad-y. I have to repeat, I only picked pi because it's a good candidate to make the point that numbers are abstract and not physical. I could have made the exact same point with 3, but people have a harder time understanding that 3 isn't any more physical than pi.
I did NOT intend to inspire any pi mysticism. There is nothing special about pi. It's just a real number. There are plenty of real numbers, many of them interesting for various reasons. It's a matter of historical contingency which ones got discovered first.
Quoting apokrisis
Yes, linear scaling factors don't matter. If you think of the number line, it doesn't matter what names we give anything. If we called 3 by the name 47, everything else would work out the same. Math wouldn't change.
* So to sum up:
- You are arguing from a computationalist point of view, but I'm not sure what point you are trying to make. Looking back I see that now. Even if I agree with you that mind is computation, there are still numbers and rocks. I possibly did not follow your argument.
- You are wrong that math is a computation. And like many computationlists, you underestimate or ignore the importance of non-computable phenomena in the world. Remember even Tegmark distinguishes between the mathematical universe hypothesis and the computable universe hypothesis. Computationalism is a very strong assumption.
* Mathematicians do math, not philosophy. My sense is that the vast majority of working mathematicians never give any thought to philosophy. When an engineer is building a bridge, do you want him spending his time contemplating the fact that there is no difference between him and the bridge? Or do you want him calculating the load factors according to state of the art engineering principles?
Well I hope some of this was on target.
Quoting fishfry
I'm definitely not claiming computationalism - or at least not Turing machine computation as you seem to suggest. The mainstream neuroscience view - since Sherrington's "enchanted loom" or Hebbs's learning networks - is some kind of neural net form of "computation".
And more to the point, it is mainstream to emphasise that the brain is involved in informational activity, not merely biochemical activity. Otherwise why is neuroscience interested in discovering the secrets of the neural code, or brain's processing architecture? It knows the biophysics of what makes a neuron fire. But how that firing then represents or symbolises something with felt meaning is the big question. And that can only be approached in terms of something other than a biochemical materialism. It demands a semiotic or information theoretic framework. Which in turn has already considered Turing computation and found it not the answer.
So broadly speaking, neuroscientists think thoughts are informational processes and not biochemical events. At the same time, they don't think the brain is literally a Turing machine or programmable computer. That might be a helpful analogy, like calling the eye a camera. But just as quickly, the caveats would begin.
Quoting fishfry
Computers are machines. They are devices that construct patterns. So yes, of course, human minds seem to operate in a fundamentally different fashion. We can grasp the whole of some pattern. We can understand it "organically" as a system of constraints, rather than as an atomistic construction.
Our abductive or intuitive approach to reasoning begins with this ability to see the whole that "stands behind" the part. We can make inferences to the best explanation. And then, having framed an axiom or hypothesis, we are also quite good at deducing consequences and confirming by observation.
So when it comes to mathematical truth, that is what we think we are doing. We notice something about the world. We then leap towards some rational principle that could "stand behind" this something as its more general constraint.
Turing machines are really bad at making such a holistic generalisation. Neural network computers are our attempt to build machines that are good at implementing this precise inferential leap.
Quoting fishfry
Yeah. I don't claim complete substrate independence. But then my "computationalism" is a semiotic or embodied one. The whole point is that it hinges on a separation which then allows an interaction.
A Turing machine does not self-replicate. A Turing machine does not have to manage its material flows or compete with other TMs. But a living thing is all about regulating its physics with information. So an independence from physical substrate (an epistemic cut) is required by life and mind. But only so as to be able to regulate that physics - bend it in the direction which is making the autopoietic wholeness that is "an organism".
Quoting fishfry
Yes, you can measure one side of the computational story in terms of entropy production. But how do you measure the other side of the story in terms of "negentropy" production? The fact that your computer runs either hotter or colder doesn't say much about whether its eventual output is righter or wronger.
Quoting fishfry
We are labouring the point. If you really can't see the difference between syntax and semantics by now, things are likely hopeless.
You keep talking about the physical events as if they are the informational processes. Of course a neuron or a transistor or a membrane receptor or a speedometer can be described in terms of their "physics". But it is hardly the level of description that explains "the process" which we are interested in.
To reduce functional or informational processes to atomistic material events becomes a nonsense. Especially for true computationalism. The only time we are interested in the physics of a logic gate is when it doesn't behave like a logic gate - that is when it has some uncontrolled physical process going on.
So algorithms are extreme mechanistic dualism in fact. You don't even have to run a programme for it to "have a result". The result could only be different if the physics of the real world somehow intruded, And then we would say the computer had a bug. It over-heated or something.
And maths is kind of like that. We imagine it as transcendent and eternal truths - things that would be true without ever needing the reality of physical instantiation. Pure information. It is crazy to talk of Euclidean maths as existing in some geezer's long dead brain.
Quoting fishfry
Why do you interpret that as a mystical statement? My point was that it is not a mystery because it is what you would expect from principles of physicalist symmetry. If every kind of difference gets cancelled (as the negatives erase the positives) then what you are left with is the mid-point balance. It would be natural to expect "flatness" as the emergent limit state.
Quoting fishfry
Well it is your choice to ignore what we know to be fundamental in preference for what we know to be emergent.
I can't agree that it makes for good metaphysics. And I think you just want to avoid having to make a better argument.
Quoting fishfry
Fine. The philosophical issue here is not the pragmatics of mathematical research. And I even agree that mathematical research - in being an informational theoretic exercise - would deliberately insulate itself from such fundamental metaphysical issues. Maths doesn't really want to even concern itself with geometry - the physical constraints of space - let alone with actual materiality, or the constraints of energy, the possibilities of change. So - as institutional habit - integers are as real as rocks.
Except they are then ... ideas? Constructs? Thoughts in the head?
You seem to want it both ways. And that winds up in Platonism.
That is why my own position is the semiotic one where the integers are the ideal limits on materiality. That is a formula of words that both accepts a strong difference and a strong connection between the two sides of the semiotic equation. Information is real if it is causal. And being an actual limit on material freedom is pretty clearly causal.
Quoting fishfry
See earlier where I spoke about abductive reasoning and our ability to make inferential leaps. Gödel validates my approach here. The failure of logical atomism is the solid ground for the holist. It is why a semiotic approach to reality is justified.
Quoting fishfry
You mentioned pi. I am just highlighting how the usual woo-woo aspect - the fact that there is just this "one number" picked at random out of all the numbers on the number-line - masks a bigger story. The woo-woo evaporates when you see there is a "material" process that picks out a value for "being flat". Two kinds of possible curvature had a mid-point balance. Pi is a number that emerges due to something more holistic going on. The fact that it emerges "right there" on the number-line is not some kind of weird magic.
It is even easier to see with other constant like e that are directly derived from growth processes. There the contrasting actions that produce the emergent ratio are in plain sight. It is funny that e should be 2.71828. But then that becomes obvious when it is realised that growth always has to start from some thing that is just itself 1. There is no reason to think of e as anything but natural after that.
Quoting fishfry
But I am not Kantian, except in a loose sense. I'm Peircean in the way Peirce fixed Kant.
And I'm arguing flatness is special as the mid-point of opposing extremes of curvature. It has physically important properties too. Only flat geometries preserve invariance under transformations of scale. That is a really important emergent property when it comes to things like Universes.
Quoting fishfry
And as I repeat, it is very important metaphysically that absolute scale invariance only appears at a particular numeric value of pi. That is how a Universe is even possible.
So you are focused on the triviality of pi being given some particular position on the number line - look guys, its 3.141592653589793238462643383279502884197169399375105820974944592307816406286 208998628034825342117067982148086513282306647093844609550582231725359408 ...
And that is what makes folk go woo. It seems both weirdly specific and weirdly random. There seems no natural reason for the value.
But it's a ratio derived from the radius being granted as the natural unit. Let's call the radius 1. Let's get a grip on this weird thing called curvature by starting with the "most natural part of the story" - a line segment. That gets to be "1" on the number-line.
Well, as I say, once mathematicians woke up to the fact that flatness was a rather special case of curvature, and once physicists in turn woke up to the fact that scale invariance was essential to any kind of workable Universe (its called rather grandly the cosmological principle), well, maybe it is the ratio that should be called "1". A straight line segment is only a natural unit in the context of an already flat space which supports unlimited scale transformations. It depends on the emergent fact of parallel lines or infinite rays being an actual possibility.
Quoting fishfry
I am being anti-mystical in pointing out the very physical basis of pi as a number. It is a ratio that picks out a critical geometric balance.
The number 3 is trivial by comparison. Well there are physical arguments for why the geometry of universes are optimal if they have just three orthogonal spatial directions. But 3 as a member of the integers has no numeric specialness by design. The special or natural numbers are 1 and 0. We see this in the symmetries captured by identity operations. There is something basic or universal when we hit the bedrock that is a symmetry or invariance.
You would call it a mystical fact perhaps. I see it as quite reasonable and self-explanatory.
Quoting fishfry
Nope. At least not your notion of computation as Turing machine/programmable computation.
I take an information theoretic perspective. And more specifically, a semiotic one. In technology terms, neural networks come the closest to implementing that notion of computation.
And numbers vs rocks is a distinction that relies on a classical metaphysics - one in which the divide between observers and observables does not present an epistemic difficulty. The epistemic cut - the necessary separation of the information from the physics - can be treated as an ontological fact.
So my positions on both "mind is a computation" and "reality is classical" are the same. Semiotics starts from the view that there is no fundamental ontic division of observers and observables. But that is also the division which must emerge via some epistemic cut. It is the basis of intelligibility. And even the Universe can only exist to the degree it hangs together in intelligible fashion.
Hence why maths tends to be unreasonably effective at describing the Universe. Or being in general.
Quoting fishfry
Labouring the point still, but I'm sorry. I'm not a computationalist in the sense you are hoping for. Indeed, that was what I was accusing you of. You seem to believe reality is a machine. An account of physical events is sufficient.
But yes, you also seem to say the opposite. This is a symptom that your metaphysics is "commonsensical" and not well thought out.
Quoting fishfry
Again, bully for mathematicians. Bully for engineers. Bully even for most physicists (as very few are employed in frontier theory construction).
But it is curious to be complaining about metaphysics where metaphysics is appropriate.
And so far you haven't put forward any clear exposition of your own epistemic position, let alone given a clear justification for it. You just hoped to be able to label me with some obviously weak ontology that I spend most of my time arguing against.
* First, there are Turing machines (TMs). The Church-Turing thesis says that anything we can compute, can be computed by a TM. This is an eighty year old core idea in computer science that has never been refuted. If tomorrow morning professor so-and-so in Helsinki publishes a paper called, "A mode of computation that's not a TM," it would rock the computer science world and it would make the popular media. "80 year old computer theory debunked," etc.
This hasn't happened. As far as anyone in the world knows, everything that we would call a computation can be implemented as a TM.
* Real-world neural nets are TMs. This must be true; if not, Church-Turing would be broken and we'd all have heard about it.
What is a neural net (NN)? It consists of a set of nodes, each assigned a numeric weight. Then we apply some logic: If the weight of this node is such and so and its immediate neighbors are such and so and their neighbors are such and so, then do something. This is perfectly conventional programming. The greatest weak AI in the world, such as AlphaGo Zero, is a conventional computer program implemented on conventional hardware. A real-world implementation of a TM.
To be sure, neural nets are very clever ways go organize a conventional computation. But they are conventional computations nonetheless.
* Theorestical neural nets. In the abstract model, the numeric weights of the nodes can be real numbers. Since in general it takes an infinite amount of information to specify a real number, there are no real-world implementations of theoretical NNs. I'm not aware of the theory of computation behind NN and how they relate to Church-Turing.
* Wetware NNs such as the brain. Since brains are physical, even if they are NN's, they are TMs. Moreover, the idea that the mind implemented by the brain is a NN is a speculative idea. Nobody has proof. There's no reason to believe a mind/brain is a TM and plenty of reasons to doubt it.
Quoting apokrisis
Any physical implmentation of a NN is a TM. If you disagree then you either believe the Church-Turing thesis has been falsified (which it hasn't) or that the brain contains nodes that can represent arbitrary real numbers (absurd) or you have some other justification for your claim. Please provide such justification.
As far as what constitutes mainstream neuroscience, I'm not qualified to judge. I hope you would agree that the opinions of neuroscientists donot constitute a refutation of Church-Turing, merely ignorance of it.
Quoting apokrisis
It may be mainstream speculation, but it is certainly not mainstream established fact. But again, why are you so hung up on the opinions of neuroscientists? Since any real-world NN must be implemented as a TM, the burden is on you to explain yourself.
Quoting apokrisis
How does that prove a real-world NN isn't a TM? You are committed to your argumentum ad populum but you don't seem to be able to reason on your own.
Quoting apokrisis
Oh, I thought it was a computer program as you keep claiming. Or a NN, which is just a particular kind of computer program. Now you admit that we DON'T understand how firing neurons give rise to mind. Well then at last we agree.
Quoting apokrisis
Why?
Quoting apokrisis
Why?
Quoting apokrisis
Funny you should say that, since many people (wrongly) believe the mind is literally a TM computation. But if the mind is ANY kind of computation, you have to explain how it could be a computation yet not be a TM. This point does not seem to be appreciated in the literature
Quoting apokrisis
Some do, some don't. Some scientists used to think heat was caused by phlogiston. What of it? You are continually trying to substitute claims about the opinions of some neuroscientists for thinking things through on your own.
Quoting apokrisis
I'm glad that you agree with me on at least this point. The problem is that there is no other mode of computation as far as we know.
Quoting apokrisis
If you agree with me why do you keep trying to disagree? I have no idea what your point is. You go back and forth on your own opinion.
Quoting apokrisis
Yes. Which neither confirms nor denies that mind is a computation, since even the weak AI's are quite impressive these days in seeing the whole, as in facial recognition.
Quoting apokrisis
You are eloquently agreeing with my point.
Quoting apokrisis
These days, strangely and counterintuitively, TMs are incredibly good at generalization and "gestalt," at least in constrained domains. AlphaGo Zero is mind-blowing in its philosophical implications and AlphaGo Zero is a TM.
Quoting apokrisis
Agreed. NN's are a clever way of organizing a conventional TM. But every NN is implemented as a TM. They're computer programs implemented on conventional hardware. Please tell me you understand this point. There are no magic NN computers. They're NN algorithms implemented on TMs.
Quoting apokrisis
If it's a computation then it's a TM. You need to deal with this point.
Quoting apokrisis
Neither does a person without children. What does that have to do with the subject at hand? Red herring.
Quoting apokrisis
Ever hear of core wars? Oldtime hackers used to write programs that would compete with each other for machine resources. Of course TMs can be programmed to compete with other TMs.
Quoting apokrisis
You are confusing the issue by bringing up living things. Nothing to do with the subject at hand.
Quoting apokrisis
I don't see why. Searle believes mind is a function of the physical brain, just not a computational one. You don't need mysticism or duality.
Quoting apokrisis
Autopoeietic. Whatever. What's that mean? I could look it up but I'd like you to explain this in your own words what your point is. NNs are TMs and if you think the mind is a computation then you think the mind is a TM. You have to deal with that by denying it (with evidence) or accepting it.
Quoting apokrisis
I don't follow the relevance of that para.
Quoting apokrisis
If you don't see the difference between fish and bicycles, things are likely hopeless. WTF? You think I don't know the difference between syntax and semantics? You're flailing.
Quoting apokrisis
No no. But informational processes ARE physical events. Running Euclid's algorithm in a supercomputer or with pencil and paper are physical processes [not events]. They require energy and output heat. The description of the algorithm, the program, does not compute anything.
Quoting apokrisis
It doesn't explain mind. It only points out that you don't need duality to explain computation. Computation is a physical process. [And not the converse as you tried to claim I said earlier].
Quoting apokrisis
Why? Who's the dualist now? What kind of mystical process are you believing in? If my mind is not a function of my physical brain, what do you think it is, exactly? Are you a dualist or not?
Quoting apokrisis
Uncontrolled physical process? You know you are not speaking coherently these past few paragraphs. You're flailing randomly.
Quoting apokrisis
Dualism, why? I program a computer to add 2 + 2, it outputs 4. Where is the dualism? The computer inputs electricity, outputs heat, and performs a computation. I really don't understand your mysticism around this very commonplace and well-understood phenomenon of computation.
Quoting apokrisis
That's just wrong. If I write down the Euclidean algorithm, it has no result. Only when I implement the program on a physical substrate and execute the algorithm does it produce a result. If you don't understand this there really is nothing to talk about.
Quoting apokrisis
What of it? You just claimed a program need not be executed to produce a result. That's "not even wrong." Its a profound misunderstanding of computation.
Quoting apokrisis
Ok we're Platonists today. Fair enough. But where do these truths live? You are quite the mystic.
Quoting apokrisis
Really? Crazy? That the best you can do in lieu of an actual argument? You haven't made a single rational argument in this entire post. I don't think you have one.
Quoting apokrisis
Nonsense. Mathematical nonsense and physical nonsense. You must have missed the Einstenian revolution. It's not 1900 anymore.
Quoting apokrisis
Ah, emergence. Another murky concept. Hydrogen's not wet and oxygen's not wet but water is wet. Zowie, cosmic.
I honestly have no idea what you are going on about. I really don't think you are making any sense at all.
Quoting apokrisis
Didn't I already remind you earlier that Gödel disproved that math is an information-theoretic exercise? Why are you doubling down on a claim I've already falsified?
Quoting apokrisis
You seem to be back in 1840, railing against the great discovery of non-Euclidean geometry. Is that your complaint? That math isn't physics? I'm sure the physicists agree with you.
Quoting apokrisis
The nature of mathematical truth is indeed an open question.
Quoting apokrisis
I just want you to say something that's reasonably on topic and that makes some sort of sense. I have no idea what you're going on about here.
Quoting apokrisis
"the integers are the ideal limits on materiality" -- This is supposed to make logical sense to me? Is this some sort of postmodern theory? I confess I don't take postmodern mathematical musings very seriously. Perhaps you do. If you would take the time to explain what you mean by "the integers are the ideal limits on materiality" then perhaps I'd learn something.
Quoting apokrisis
You sound like a raving postmodernist. Perhaps you are a postmodernist and I'm insuffiently appreciative of that point of view. That may well be the case.
Quoting apokrisis
Semiotic. Whatever. Explain yourself clearly if you can. Can you?
Quoting apokrisis
Tell me something. What is the true value of 3? And why does it emerge "right there" on the number line, halfway between 2 and 4?
Quoting apokrisis
It's hilarious.
Quoting apokrisis
Mystical word salad.
Quoting apokrisis
Maybe we better not go there. You remember how that went last time. Although I suspect this is the problem. You are arguing from a very particular point of view. But you are not willing or able to explain yourself. So everything that anyone says is wrong from your point of view, and you're right, and you're condescending, but you can't explain your ideas in everyday English. And in my experience, you don't really understand a lot of the ideas whose terminology you carelessly sling around.
Quoting apokrisis
Oh to be back in 1840 when Euclidian geometry was given to us by God.
Quoting apokrisis
What is the specific numeric value of 3? I already explained to you that pi is a particular real number whose value does not change and has nothing to do with geometry.
Quoting apokrisis
What is the natural reason for the value of 3?
Quoting apokrisis
Bearing in mind that the unit distance is arbitrary. A linear scaling factor would make no difference.
Quoting apokrisis
I have no doubt that what you're saying makes perfect sense to you. It makes no sense to me. I don't get where you're coming from.
Quoting apokrisis
Even if I grant your point, what of it? Why are you going on about pi?
Quoting apokrisis
Then explain to me what mystical geometric balance causes 3 to have the exact value that is has.
Quoting apokrisis
Look, maybe Pearce said all this. I haven't the knowledge to comment. You seem to know a lot about this, or at least you know the buzzwords. It's pointless to argue with you about it.
Quoting apokrisis
I can no longer converse with you on this point.
Quoting apokrisis
What is your model of computation? And how do you square it with the Church-Turing thesis?
Quoting apokrisis
Yes and any NN that can exist in the physical world is a TM. You have to refute this or accept it. I'm perfectly willing to be shown wrong, since I'd learn something. Do it.
Quoting apokrisis
You are the buzzword king. You do have a hard time translating your buzzwords into ideas that you can explain to people.
Quoting apokrisis
If mind is a computation and if it's implemented on a physical brain, then it's a TM.
Quoting apokrisis
I yield to your facility with buzzwords.
Quoting apokrisis
It's a puzzler alright. Or perhaps math is only telling us something about our own minds, and not the universe at large. That's a possibility too.
Quoting apokrisis
Well we're back to the NN = TM issue again. That's the core issue here. You think there's a mode of computation that is not a TM and that can be physically implemented in the brain. This I deny. Computer science is on my side I believe. I could be wrong. I await clarity.
Quoting apokrisis
I have never believed that and I strenuously oppose it. But it's normal for you to think I've said the opposite of what I think I said. We may just have to live with that.
Quoting apokrisis
I admit I'm not much of a philosopher. If you ever took the trouble to explain your points of view to me, I'd learn something. But that never seems to happen.
Quoting apokrisis
You were complaining that mathematicians aren't in a state of "epistemic shock." I pointed out that when mathematicians are doing math, they're not doing philosophy. You want people to do their jobs, not get lost in the wonder of it all.
Quoting apokrisis
I haven't got much of an epistemic position today. I vaguely recall originally making some minor point which I no longer remember. I don't believe the mind is a TM and I don't believe real-world NN's are anything other than TMs. That's plenty of position. By the way don't you mean ontic position? What is, rather than how we know what is? I'm even confused about your nonstandard use of "epistemic." I never understand anything you say.
Quoting apokrisis
LOL. And you are doing the same to me.
Bottom line, why don't you just explain to me why you think a real-world NN is anything other than a TM. That's at least one subject that might be of interest, and it's one where we're reasonably on the same page even though we have different opinions.
Quoting fishfry
Hmm. So what I have got from this exchange is that you struggle to keep track of your own arguments because you don't actually have a well constructed metaphysical position. And when you encounter someone who does, you bluster and ad hom. Nice.
And so here now you have diverted the discussion to something that you hope might be safe ground.
I said that mainstream neuroscience would reject the reductive materialist notion that abstract thoughts are just biochemical processes in the brain. In some fashion - still not fully understood of course - they would be considered informational and semiotic processes.
You then leapt to the idea that this meant the activities of the brain are computational processes - Turing machine computational.
I replied no, a TM is a dualistic device. The software is absolutely divorced from the world which gives it rule-bound play any material meaning. It is presumed that the hardware supporting the action has no entropic cost. It is presumed that the input and the outputs of this finite state machine are meaningful to some further intelligence outside it. So a TM is just a syntactic device. It can blindly follow rules. But at no point in its mathematical-strength definition is there any semantics included.
And then, so far as neuroscientists would consider the brain some kind of computer, it would be like a neural network. Which is different from neuroscientists thinking the brain IS a neural network. Rather, it is neural networks which are like a semiotic relation.
Neural networks are meant to learn from the world by experience. They don't have a programming language and so they don't have a set of syntactic tokens to shuffle about according to some set of computational grammar. And while they can of course emulate a Turing Machine - just like we can emulate a TM too - that doesn't mean they are TMs. It just means they can follow rules that shuffle symbols without needing to understand anything about what they are doing. Semantics is optional to blind programmatic rule following.
So you made a wild claim - thoughts are nothing more than biochemistry. Now you want to defend the opposite thesis - thoughts are nothing more than Turing computation. Or no, you realise that is ridiculous. So you want to pretend that is my position instead.
Quoting fishfry
Then you don't seem to be interested in metaphysics even as it touches on the reality of numbers. It appears largely that you reject what physicalism might have to say about "reality" just because looking up "buzzwords" is such a tiresome chore ... when you already have all the answers.
I was hoping that focusing on the reality of mathematical constants might have got us somewhere. Yet it appears you haven't even really thought about the reason constants emerge as limits on material action in physical systems. So that was a waste of time too.
Oh well. I was expecting too much, obviously.
I'd love to chat with you without the snark. This isn't fun. I'm going to withdraw. Suggest you investigate the relationship between neural nets and Turing machines. I did a little research since my last post and apparently neural nets are computationally weaker than TMs. Some neural nets are Turing complete but some aren't. And of course none have computational abilities beyond the TM, since we know of no such thing in the physical world.
All the best.
And I have investigated neural nets. Your posts on the issue reveal you haven’t really.
Then you must have a substantive response to my point that if mind is an informational function of a neural net executing in the brain, then it can be implemented as a Turing machine. So it's not logically consistent to claim that mind is a neural net without admitting that your position requires you to also agree that mind must be a TM.
That is:
* If you claim that mind is a neural net; then you must also agree that mind is a TM.
I could well be wrong. If you have a substantive response to my point then perhaps I'll learn something. If you don't, you don't.
Either you understand the difference between emulating a TM and being a TM, or you don't. Either you understand the difference between analog computers and digital computers, or you don't. Either you understand the difference between semantics and syntax, or you don't. Etc, etc.
Maybe you could start with this famous philosophy of mind argument - https://en.wikipedia.org/wiki/Chinese_room#Chinese_room_and_Turing_completeness
To understand my biosemiotic take on the issue, this is a nice foundational paper - http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.18.1316&rep=rep1&type=pdf
Are you actually incapable of making a substantive argument? "You are wrong because Point 1, Point 2, and Point 3?" You simply can't do it?
Make a substantive argument or show everyone on this site that you're not capable of doing so.
Are you actually making the claim that even though a NN can be emulated by a TM, the NN somehow implements semantics? That's so wrong ... well it's wrong. Leave it at that. A thing that can be implemented or emulated as an algorithm is an algorithm. But if that's your argument, so be it. But you haven't provided an argument. You do not possess the ability to outline a rational argument. You make a claim but you provide no substantive argument.
And for what it's worth, you completely miss the point of the CRA. If you think an NN that can be emulated by a TM can nevertheless embody semantics, you simply have no idea what the CRA says.
Nope. I made the point that humans and NNs can emulate TMs. (You did claim to be familiar with the CRA?) However that doesn't make either of them TMs.
I also said neuroscientists find NNs to be biologically realistic models of neural processes. There is no reason to think brains are finite state automata. There is no reason to think they are programmable computers (von Neumann machines). There is no reason to think they are Turing complete. But - given that NNs are inspired by the biology - it is not much of a surprise that NNs implemented even as logic devices show some of the important functionality we associate with nervous systems.
So NNs are goodmodels. TMs, by contrast, are woeful models of brain function.
Can an NN have semantics or is it also just a syntactic device? Well, it all rather depends now on how you define semantics. And that is what biosemiotics concerns itself with. One would need a general physicalist theory of semantics to answer the question in some quantitative fashion.
I would say the NNs built to date aren't really semantic. They are just pattern matching systems. And they require supervised learning, so the semantics are clearly "in the mind" of their human trainers. But arguably they are getting near the abilities of an ant or cockroach.
I would say that is still only in terms of pattern matching ability. An embodied view of cognition would say that a hell of a lot is still missing in terms of an actual ability to "makes sense of the world" even at that level. NN designers haven't even got their heads around the kind of functionality they need to start implementing as the "learning algorithms" on that score.
I could say a lot more about the semantic issue, but it's way off topic for this thread.
The issue was whether maths is Platonically real or a free creation of the human mind. I argued for a third position - one which says the maths that is "unreasonably effective" when it comes to physicalist theories, is so because it describes real physicalist limits on reality.
So enough of the sideshow. You only turn anything I say back to front anyway.
Numbers are real, and establish causal relationships, and exist in space and time. Numbers exist as little scribbles or sounds that refer to a quantity. Numbers exist in space and time as little scribbles and sounds, and also the quantity they refer to exist in space and time. Some number has a different causal effect than some other number. If you need 100 points to win a game, you keep playing until you get 100 points, at which point you stop playing and start a new game.
Oh I quite agree with that. I can take a pencil and paper and step through a program. It's a common debugging technique. And you're right, that doesn't make me a TM. But I can do many things that CAN'T be emulated by a TM. Like understand Chinese. If I could understand Chinese. That's one of Searle's points that he's made over the years. When he speaks English he's doing something very different in character from what he's doing when he speaks Chinese using the symbolic rules.
Quoting apokrisis
Peace brother. I'm out.
Err, yeah. As I was saying.
But the problem was you began this by claiming biochemistry is capable of things like understanding Chinese. :)
That's my understanding of Searle's position as well. The mechanism remains to be discovered.
It is well know that Searle fluffs around the issue because he has some broke-arse property dualism in mind.
But he usually talks about neural processes and brain structures as the likely level where first person experience might "pop out" into existence as an emergent property of a third person material world.
I've not seen him make a positive assertion that consciousness would be emergent just from "biochemistry", sans all that rather suggestive neural circuitry. So you might want to check your understanding.
Peace out, as they used to say.
Now that you mention it, I can narrow down my claim. There are really two issues here: What I think about mind; and what I think Searle thinks about mind.
My belief in the correctness of what I think is in no way affected by the state of my knowledge about what Searle thinks.
My thesis here is that mind arises from a physical process in the brain; but that it is not a computational process in any way that we currently understand computation. It's not a TM or an NN or a cellular automata or anything else along those lines.
I don't know what the actual mechanism might be. I'm not sure even what it means to say that there is a process that follows physical law but that is not a computation. I think these are matters for future geniuses to work out. I think this will take another revolution in physics.
Now as it happens, the way I got my opinion is that I read something Searle wrote, or perhaps said in a video. I may well have misunderstood what he said; but in any event it sparked this thesis in my mind, or gave clarity to some vaguer notions I'd been having.
So Searle is the inspiration for my opinion but may not himself actually share my opinion. But really -- isn't that classic Searle? For decades people have been misunderstanding the CRA yet have been intellectually inspired by their own misunderstandings.
ps -- Oh wait you said necessary and sufficient. No I don't think biochemistry is necessary. Or sufficient. It just "happens to be the case" in this instance. It's possible that machinery might become conscious, so biochemistry's not necessary. And there's plenty of biochemical matter walking around that's not particularly conscious, so biochemistry is not sufficient.
It seems curious that it was only just a few posts back that you were trumpeting the mind-like abilities of NNs. So if they were inspired by the "computational" structure of the brain, it is surprising they should indeed be so effective at machine learning, and yet the brain itself would not function along these lines.
Quoting fishfry
Sounds legit.
Quoting fishfry
So that is a retraction of your original statement coupled to a backtrack on the retraction?
It is the structure of the matter that matters and not the particular matter. But you don't want to say the structure implements any kind of informational process?
I oppose that notion every chance I get. It's not possible that I expressed such an opinion. Two possibilities:
* You misunderstood something I wrote; or
* I expressed myself so badly that I communicated the opposite of what I intended.
One's just as likely as the other.
But no, NN's are not "mind-like." It's starting to become my mission in life to explain to people why NN's are *NOT* "mind-like." To the extent that I fail to make a convincing argument, I need to work on my argument. To the extent that I'm giving you the opposite impression of what I'm actually trying to say, I have to work harder not to do that.
Quoting apokrisis
Airplanes are stunningly effective at flying, yet birds don't work that way.
Quoting apokrisis
You agree with me that perhaps the explanation of mind must await the next revolution (or two) in physics? If you agreed with that point you'd be halfway to agreeing with the rest of my thesis. There's a lot we don't know.
Quoting apokrisis
You asked me if I thought biochemistry was both necessary and sufficient for mind. I said it's neither necessary nor sufficient. You are seeing this as a retraction? How so?
Quoting apokrisis
I don't know. Perhaps it has to be biological. Perhaps not. I don't think it's relevant to my argument. Whatever mind is, it's not a computation. But I take no position on whether it has to be biological or not. Is that more clear? I don't know what you think I'm retracting.
Quoting apokrisis
Hmmm ... that's kind of an interesting technical question. So there's the neural wetware of the brain, and you are asking me if it is possible that SOME informational process is implemented.
Um ... well ... sure. Why not. If I blink my eyes at you in morse code I'm digitizing my thoughts. For that matter, I can execute the Euclidean algorithm with pencil and paper. So yes, wetware can certainly implement computational processes. But not everything wetware does can be explained by a computation.
You agree with Schopenhauer:
[i]Thus also, whoever has recognised the law of causation, the
aspect of the principle of sufficient reason which appears in what
fills these forms (space and time) as objects of perception, that
is to say matter, has completely mastered the nature of matter as
such, for matter is nothing more than causation, as any one will
see at once if he reflects. Its true being is its action, nor can we
possibly conceive it as having any other meaning. Only as active
does it fill space and time; its action upon the immediate object
(which is itself matter) determines that perception in which alone
it exists. The consequence of the action of any material object
upon any other, is known only in so far as the latter acts upon the
immediate object in a different way from that in which it acted
before; it consists only of this. Cause and effect thus constitute
the whole nature of matter; its true being is its action.[/i]
Fine. I would agree that NNs are not biologically realistic in some fundamental ways. But also, NNs are an attempt to be more biologically realistic in some important structural or information-processing fashion.
So this could easily be an argument over whether the glass is half full or half empty. That is why the epistemology of NNs demands especial care in a Philosophy of Mind discussion.
Quoting fishfry
But what is the "unreasonably effective" feature they share? Is it an aerofoil wing that creates lift?
I agree that human machines are just basically different from biological organisms. However again, you need some actual general metaphysical argument to spell out the precise nature of that difference. And that is what I'm talking about with biosemiosis, autopoiesis and other "buzzwords".
You need a theory of the distinction if you want to say anything definite on the matter. And you seem quite dismissive of the literature here.
Quoting fishfry
No. I was being sarcastic.
Physics is already undergoing the right kinds of revolution anyway. Thermodynamics is becoming foundational. Physics is becoming information theoretic. Holism and emergence can now be modelled in a variety of ways.
So Newtonian materialism is out-dated. Existence can be understood as a dissipative process. And that is a framework which biology and neurology slot straight into.
Quoting fishfry
Well I would say this shows you don't have an appropriate general metaphysical framework. It has to be a central issue if you are arguing either for or against artificial life and mind.
That is why I urged you to read that Pattee paper.
Quoting fishfry
That's a hand-waving statement, so not much use in a serious debate here.
At the moment I have no clue what you even mean by "mind". I get the impression it is probably the standard dualistic substance ontology - a sensing stuff, a bunch of "feels".
So we wouldn't even be on the same page for a serious discussion in terms of a comparison of neurological processes and computational mechanisms. You are likely already convinced that there is no physicalist understanding of what brains do.
Quoting fishfry
You seem to entirely miss the point.
You appear to believe that TMs completely define all possible notions of computation, information and semiosis. And so any question about "information processes" or "processing architecture" gets immediately translated into a TM view.
But just maybe TMs are a very tiny fragment of a much larger landscape.
Of course, there is something immensely powerful about TMs in being (almost) pure syntax/no semantics. In short, they are (near) perfect machines. They represent a completely constrained and rule-bound universe. And so they leave out all the "messiness" of the physical and biological world. They leave out, in fact, information as traditionally understood - ie: information as meaning.
It is like the syntax of Boolean logic. To reconnect to the OP, there is something "unreasonably effective" about reaching the limits on a de-semanticised view of reality - one where we just model reality in terms of its simplest syntactical rules.
So TMs and Boolean logic idealise reality. They abstract away the materiality or particularity of physicalist semantics to arrive at the simplest, sparest, syntactical forms.
Great. Defining the ultimate limits of reality is what it is all about. But maybe there is such a thing as over-simplification.
Machines are rule-bound artificial systems. And so they can't construct themselves. They can't give themselves purposes , they don't have autonomy. Machines are useful to us humans as it is we who get to design the machines, build them to serve some purpose.
However organisms are systems with evolved designs and purposes. They have an irreducible causal complexity. And that is their "secret". There is always semantics - or semiosis - involved.
So the whole mechanical paradigm of nature is flawed at root if it excludes the basic causal complexity of real living and minding creatures.
We can see that TMs and Boolean logic leave out formal and final cause. Well they leave out material cause as well. All they are is pure syntax. They can be used - by an organism with a purpose and a design - to represent a formal system of entailment. They can capture the description of a syntactic structure. But being such a rarified representation of reality, the computational patterns that result have an extreme real-world brittleness.
In practice, any computer program or computer circuit is incredibly prone to bugs. Just one broken link and the whole finite state automata grinds to a halt.
Organisms by contrasts not only thrive on physical instability, their very existence depends on it. Life and mind arise on the "edge of chaos" as where things are perched on the verge of falling apart, that is where the slightest extra informational nudge can push them instead into falling together.
So life and mind thrive on material dynamism. TMs and other machines only flourish where all the uncertainties of the real world have been managed out of existence by their human designers. Mindless routine following becomes possible where minds have made that a safe thing to do.
Anyway, my point is that any biologist or neurologist would understand that computers and organisms are different in this fundamental way. There is a reason why TMs are both such "universal" machines, and also the most biologically helpless of physical structures.
There is a general metaphysical paradigm that accounts for why brains aren't computers, and yet also, we could build computers that start to have some of that biological realism designed into them.
A "true" NN has to learn for itself. That's both its advantage and disadvantage. It is essentially a black box to its human owner.
I know a "mad genius" who has developed one of the currently most advanced neural network computers in the world. It runs his company for him. But he has no clue how it works inside. It grew its own "programme". And if it failed, he couldn't transfer its software to another hardware rack. He can't even do a memory back-up as such.
But because the memory doesn't work like a traditional TM device, and instead is more like a brain, that is not such a problem as it has natural fault tolerance. The failure of individual links can't corrupt the whole system.
So yep, the whole NN issue isn't clear-cut. But the field has a history now. Computer science has been exploring the degree to which neurologically realistic architectures can lead to a more organismic notion of a machine.
We already have a mathematical definition of the most non-organismic one - a TM/Boolean one - as the theoretical limit of a machine that is all syntax, no semantics. So the next question for the engineers is how to start building back in some useful biological realism. And that in turn demands a general metaphysical theory about how to define "semantic processing", or semiosis.
Then the other half of the causal story is the global form which constrains actions to locations and directions.
To reconnect to the OP, that is why I would be a realist about global constraints as well as local degrees of freedom. The two together make for a reality that has an observable structure and regularity.
Yes, Schopenhuaer's prose is clear enough, but just a little dated and pompous in tone. ;)
For Schopenhauer, the other half of the story is Will, understood as the primal urge (he didn't want to reduce it to physical forces for they are causal conditions of the manifestation itself) that we see manifested as gravity, mechanical action, chemical reaction, organic growth, hunger, sexual desire, instinct and so on. For Schopenhauer this primal urge or striving is completely devoid of any characteristics that could be understood as being possessed by anything in the world as idea.
So, no idea of time, space, causality, differentiation and so on can be coherently applied to Will. The emergence of the world as idea would be, in your terms. the symmetry breaking. The world as will is a virtual world; the thing in itself (note, it cannot be things in themselves for Schopenhuaer and their can be no causal relation between the thing in itself and the world as idea; the latter just is the former as manifest). According to Schopenhauer we can 'feel' the world as will in our experience of our own bodies. I think Peirce has a similar notion of the experience of "firtsness", but maybe I have misunderstood.
Quoting Janus
This is the problem for me. Peirce makes sense of causality as the development of reasonable habits. I can follow that as an intelligible metaphysics.
But not Schopenhauer. In the end, I can't piece together a logical description of a coming into being as a concrete self-organising process. The bits don't fit together.
Peirce liked Schelling. I can see why.
Peirce at first disliked Hegel but then came to appreciate him. Again, I see why.
Peirce seems to have been silent on Schopenhauer. Perhaps Schop just wasn't systematic enough for there to be a real metaphysical thesis to critique?
I wonder, though, whether Peirce can make sense of the development of reasonable habits in terms of something more fundamental?
Schop could say that Will comes to manifest in ever more habitual ways, which become the more reasonable as the world as idea unfolds; he could say that Will gains its increase by establishing habitual manifestations.
But isn't that the problem? The way you phrase it suggests that you have certain beliefs about the nature of fundamentality.
The semiotic view is that tychism - chance or spontaneity - is the most fundamental starting point because it has the least regularity or stability. It is the least concrete possible state. So it is not a "something" - some more basic level of substance. It is a state of unfettered anythingness. It is pure instability without habit or regulation.
This then means reality arises by a restriction on a fundamental anythingness. So Peirce has a metaphysics we can recognise from Anaximander. And one that also is now straight out of modern quantum physics and thermodynamics.
So any metaphysics that tries to get something from nothing does have a problem. But reducing everything to just something is easy, by contrast.
And because we know something does indeed exist - us and our cosmos - we already know that nothingness couldn't have been the case. So whatever our metaphysical reasoning leads us to as the "primal condition" has to be the best answer we are going to get. Which is why we would believe in Firstness, vagueness, the Apeiron, a quantum foam, or whatever best represents a condition of chaotic symmetry, a realm of utterly unstable fluctuation.
This foment may indeed sound a little like a primal raging will to exist. But does any connection to Schop go deeper than that?
Quoting Janus
Yes. Maybe Schop could be mapped to this kind of "anythingness" based metaphysics. I mean it is the general alternative option that runs through all creation stories.
Either there was nothing, and existence got created. Or else existence is a result of a disorganised everythingness that got regulated.
I just think that Peirce developed the best account of the mechanism for a self-organising everythingness. He realised it had to be a triadic tale, not merely a dualistic one.
For me, when you describe Schop's Will, it seems to be trying to stand for two things at once - both the material spontaneity and the formal constraints. It is the primal source of the energy and also the end towards which that energy is directed.
Again, a triadic view allows everything to arise emergently. It stands against the usual view where something can only come from something - the view that presumes substance to be a conserved quantity in the process of creation. Peirce's metaphysics is an open systems view - one which starts in the unlimited and develops its concreteness through self-bounding or self-closure.
So the question about what is "fundamental" is flipped. Any beginning - and any ending - have to be the least concrete kinds of causes imaginable. As the concrete is what arises in the middle between them.
That means the beginning is a Firstness or vagueness. Just pure fluctuation. And the ending is Thirdness or generality. Just the fixity of "a habit". A state where all differences are assimilated to a common idea.
So, as I say, the beginning and the end are real (ie: not nominalist). But they are logically opposed (one being vague, the other general) and both are arrived at as being as "insubstantial" as can be imagined.
The Will, by contrast, seems to exist as an efficient cause that drives the action. It is definite at the start, and gets to where it alway intended by the end.
It just lacks the formal dichotomous division of the vague and the general (as that to which the principle of non-contradiction and the law of the excluded middle respectively fail to apply). And it lacks the insubstantiality that can then stand as the contrast to the actuality, the particularity of secondness, that arises emergently "in the middle" - in good old hylomorphic fashion.
So Peirce has deep roots that are sunk right into logic itself - the laws of thought. It connects in direct fashion to Aristotelian metaphysics as well.
I don't get any of this kind of rigour from Schop. But then I've never looked into him that deeply.
Then we are fundamentally in agreement.
Quoting apokrisis
Of course. I hope you don't think I'm dismissive of the amazing achievements of weak AI these days. From chess to Go to driving cars to facial recognition.
But isn't it true that just because we can cleverly simulate an approximation of certain aspects of the human mind, that this does not necessarily mean that this is literally how the human mind works?
In other words we've invented flying machines; but that doesn't mean we've discovered the mechanism by which birds fly. On the contrary, we've shown that the way humans make flying machines is different than the way God (or nature) does. And by analogy, perhaps the way humans make thinking machines is different than the way nature does. In which case, NN's can be as clever as you like, but minds are still not NNs. Just like birds aren't jet engines with fixed wings and small bags of salted peanuts.
Quoting apokrisis
Humans can build machines that emulate what nature does, but we do it differently than nature does.
Quoting apokrisis
Can you explain what the epistemology of NNs means? That went over my head.
Quoting apokrisis
Well that's the point. Just because we can make flying machines doesn't mean that nature's flying machines work the same way. And just because we can make thinking machines doesn't mean nature's thinking machines work that way.
Quoting apokrisis
Well we have to be careful here. If I'm making a physicalist (but not computationalist) argument, then I must admit that we are machines. But we are not computations. I'm making that distinction. We are machines that think, but we are not computations.
Quoting apokrisis
My complaint about the buzzwords is that you often fail to explain your ideas to me in ways I can understand. And even if I went back to school and got a Ph.D. in philosophy, I might still miss out on Peirce. So if you care to explain your viewpoints I'd learn something.
Quoting apokrisis
No, only dismissive of your unwillingness to explain things that you know and I don't.
Quoting apokrisis
Ah, a crushing disappointment then.
Quoting apokrisis
So physics is already pretty much complete, and all we need to do is fill in the details? That's exactly what they thought in 1900, right before the quantum revolution. Surely history teaches us that there are radical paradigm shifts in physics every couple of centuries. The next one might not happen in our lifetimes but it's certain that it will happen sooner or later.
Quoting apokrisis
Just as our current ideas may soon be equally as out-dated.
Quoting apokrisis
What does that mean? I confess to having lived a dissipated life but I don't think that's what you mean.
Quoting apokrisis
Sure. No disagreement.
Quoting apokrisis
To my eternal shame. Must one have a general metaphysical framework in order to have an opinion about whether minds are algorithms? I don't know much about philosophy but I do know a lot about algorithms, and I know that algorithms are literally dumber than rocks. A rock sitting there on the ground has more claim to be a thinking entity than an algorithm does. I don't need to have a general metaphysical framework to know that.
Quoting apokrisis
If you are arguing that before I can express an opinion I need to know more about philosophy, this forum would be a lonely place if you applied that standard to everyone. But I do take your point. I'm ignorant of philosophy. I've stated that many times. I don't pretend to know what I don't know.
Quoting apokrisis
Sorry I must have missed that. Link again please?
Quoting apokrisis
It's a summary of my strongly held belief, based on a lifetime of having a mind (and occasionally losing it) and an adult lifetime of working with algorithms. Algorithms are much dumber than minds.
At the moment I have
Quoting apokrisis
Well if you're going to take me down the rabbit hole of the philosophy of mind, I'm sure you know more than I do. But I don't think that my subjective experiences and ability to be in the world as an intelligent human being are the result of an algorithmic process.
Quoting apokrisis
How can you say that? Of course there is some physicalist understanding of what brains do, even though our current state of knowledge is quite limited. And since I've said repeatedly that mind [whatever it is] is a function of brain/biochemistry, it follows that there may someday be understanding of it. Why would you think I've said the opposite?
Quoting apokrisis
I answered the literal question you asked. You should ask a more precise question then.
Quoting apokrisis
That's my understanding of the Church-Turing thesis. If you have a different idea I'd be interested to hear it.
Quoting apokrisis
I confess I don't understand the meaning of that word, and looking it up from time to time doesn't seem to help.
Quoting apokrisis
That's Church-Turing, and nobody has figured out how to defined a model of computation or information processing that violates it by not being a TM. You are very casually ignoring this point and I'd like to you to comment on it.
Quoting apokrisis
YES!! But not according to contemporary theory, because we know of no such larger computational landscape. That's exactly why I think we need a revolution in physics that shows us how to go past TMs into some mode of computation that is more powerful than a TM.
Quoting apokrisis
Yes. You are eloquently expressing my very point.
Quoting apokrisis
Aha. Here we have a point of divergence in our worldviews. Information processing is what a TM does, by definition. Flipping bits according to an algorithm.
When you say "information is meaning," that's something I absolutely deny by my definition of information. A bitstring carries information but it does not carry meaning. Only a human can say what the computation "means." Searle's intensionality. The Chinese room manipulates information but it has no notion of what any of it means. That's the entire point. One with which you sometimes seem to agree.
I don't think you can claim that information is meaning. Information is meaningless. Humans give meaning to information. Isn't that true? If I say I saw a "cat," the symbols by themselves convey know meaning. It's humans, English-speaking ones at that, who say that the word cat stands for a furry domesticated mammal that's not a dog.
Quoting apokrisis
Well you can't model all of reality with algorithms. Only certain aspects of it. The map is not the territory and the model is not the thing itself.
Quoting apokrisis
Yes. We're 100% in agreement here. Algorithms abstract certain aspects of reality.
Quoting apokrisis
Like claiming that because NNs can do clever tricks like playing Go, that the human mind must be an NN. Now THAT is over simplification.
Quoting apokrisis
If by machines you mean computations. If I am making a physicalist argument I have to claim that humans are machines but not computations.
Quoting apokrisis
Yes. Humans provide the meaning. Information by itself has no meaning. You are agreeing with me again. Been happening a lot lately!!
Quoting apokrisis
YES!!!!!!!!! That's what I'm saying. Minds go way beyond algorithms. You are totally agreeing with me.
Quoting apokrisis
Ah. No.The computational paradigm of nature is flawed. We need a breakthrough in figuring out physical explanations how meat machines like us can have minds yet not be computations. We need a revolution in physics, we need to break through the Church-Turing constraint, we need to figure out the nature of a machine that does more than computing.
Quoting apokrisis
Right. Humans aren't computations. After all this you're agreeing with me.
But then you say well yes humans aren't TMs but they are NN's. And you won't come to terms with the fact that NNs are a special case of TMs. NNs are algorithms. So you aren't gaining anything by claiming that humans are NNs and not TMs. We keep going over this point.
Quoting apokrisis
Humans too. Disease, death. All machines are imperfect. The solar system probably isn't stable. [Open question at the moment]. The universe will someday collapse or else expand into eternal cold. Reality is highly imperfect. What point are you making?
Quoting apokrisis
Well you're arguing that humans aren't computations, which I've been saying for several posts now. You agree with me. I'm gratified.
But if you want to claim that humans are NN's, you have no argument because NN's are TMs.
Quoting apokrisis
So minds aren't algorithms. You are making my point for me.
Quoting apokrisis
Right. Right. Right. Right. So the question is: What exactly are we doing that goes beyond mere algorithms? That's the question. You haven't got an answer. I haven't either. But we agree that humans are not computations. And that NNs are computations. So humans aren't NNs either. We don't understand the mechanism by which humans operate in the world.
Quoting apokrisis
Sure, the chess playing and Go playing and automobile driving and face recognizing algorithms are very impressive these days. That doesn't tell us anything about minds.
Quoting apokrisis
So you are distinguishing between the fake NN's that are merely reorganized ways of implementing TMs' and "true" NNs that are some theoretical construct that are NOT the NNs of current theory. That's it right there. You admit that you are not talking about NNs as currently understood. You are using "NN" to mean whatever it is that humans do, that's not a computation.
Quoting apokrisis
I call bs on that. Not that you don't know some guy, but that he can't back up his system. If it's built out of processors and memory devices then he can back them up just fine with perfectly conventional techniques. There is no magic hardware paradigm. There are only different ways of organizing the bit-flipping activity.
Quoting apokrisis
I spent years working on fault-tolerant systems. I think your friend is messing with you. Error-correcting codes, fault-tolerance via hardware redundancy, via voting, via consensus algorithms, are all well-studied ideas since the 1970's. Cryptocurrencies fall into that space, they're a brilliant solution to the problem of distributed consensus on adversarial networks.
Your friend doesn't have a computer that's "more like a brain." He's just taking advantage of his philosophically trained friend who thinks computers are magic, and funnin' ya.
Quoting apokrisis
Absolutely. But they have not broken the Church-Turing barrier. They have not implemented a mind. And they have not by any stretch of the imagination proved that minds are NNs, any more than a 747 proves that a bird is like an airplane.
Quoting apokrisis
Ok, you're arguing that the AI community needs to learn more philosophy. A traditional argument. But it still doesn't mean that minds are algorithms or NNs. It only means that NNs are better at building thinking machines than earlier computing paradigms.
I do hope you agree that building artificial machines that exhibit "thinking" in constrained domains is one thing; and that claiming that the human mind works that same way is quite another.
This ignores the fact that the flying machine designers quickly gave up trying to copy the flapping wings of birds and instead focused on a non-bird model of flying machines. The flapping did not prove "unreasonably effective".
Whereas the opposite is the case with NNs. Having got programmable computers, it was the case that even just emulating biologically-inspired information processing architectures was "unreasonably effective" for certain tasks, like pattern matching.
So that is a particularly inapt comparison with which to make your case.
Quoting fishfry
So an organism is a machine? You seem out of touch with biology.
Quoting fishfry
Artificial Life Needs a Real Epistemology - H. H. Pattee
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.18.1316&rep=rep1&type=pdf
Quoting fishfry
So on the one hand you can't even define what you might mean by mind. On the other, you can make confident claims about neuroscience having a quite limited understanding. And you keep reverting to talk of "brain biochemistry" when the question is about cognitive functions.
Don't you see the inconsistency of one minute admitting to knowing little, the next to be making a sweeping judgement of the whole field?
Quoting fishfry
That defines computation in the general limit ... if you are computing number theoretic functions.
So perhaps brains might not be that kind of "computer". Maybe there is not a single arithmetic operation involved in their neural processes. Maybe even "summing weights" is just an analogy for the integrative processes of brain cells. Church-Turing may have zilch to do with neurology. And yet it is still wrong to then attribute neural information processes to "biochemistry".
And how could you have a view either way without a little more neuroscience to inform your opinion?
Quoting fishfry
Given that TMs require no more physics than a gate that can read, write and erase a symbol on an infinite tape, why the heck would we expect new physics to make a difference to Turing universal computation?
The power of Turing machines is that they need the least physics we can imagine. What more do you want - time travel, Hilbert space, quantum teleportation? That's back to front. It is the virtual elimination of any complicated physics which is the guarantee of the computational universality.
Quoting fishfry
Who could win an argument against your private definitions?
So let's stick to the real world of science, maths and philosophy. If you want to talk about Shannon entropy, fine. But then we all know that is based on counting meaningless bits. If we understood the pattern to mean something, then each successive bit would fail to be such a surprise.
If I know you are transmitting the digits of pi, I could stop you right after you said "3".
Quoting fishfry
You don't get it. Information theory defines a baseline where the meaning of a bit string is maximally uncertain. Each bit says nothing about the following bit. Then from that baseline, you can start to quantify the semantics. You can derive measures such as mutual information that speak to the information content.
Quoting fishfry
That's one of the advantages of a semiotic approach to the whole issue. It recognises that there is a modelling relation involved. A symbol has meaning due to a habit of interpretation. That habit is tied to action in the world. So the informational side of the equation is causally connected to the material side. There is only meaning in relation to the material consequences of any beliefs.
Again, read Pattee - http://www.academia.edu/3144895/The_Necessity_of_Biosemiotics_Matter-Symbol_Complementarity
Quoting fishfry
You keep misrepresenting my argument.
The significance of an NN would be that it captures something important about brain cognition. That is different from claiming the brain is literally just an NN.
And you seem confused about algorithms. They are rules for making calculations. So they are something we think it meaningful for a TM to do. They are not the barest syntax of rule following we can imagine. They are semantic actions performed on a machine.
So already we are into the real world where computation carries extra semantic baggage. The algorithms are intended to represent some actual informational process. This could be just handling a company's payroll or driving a video display. Or it could be an attempt to mimic the connective behaviour of neural circuits.
A TM is just a universal algorithm runner. How we then exploit that is down to the kind of information processing we think might be meaningful. We have to write an algorithm that seems to perform the task we have in mind. That could be representing brain functions. It could be representing accounting functions or moving image functions. Universal Turing machines have zilch to say about whether we humans are choosing to run usefully realistic routines or just scrambled garbage randomly concocted.
You are confusing yourself in jumping so interchangeably between talk of TMs, information, computation and algorithms.
Quoting fishfry
Again, we write the algorithms. They have zilch to do with the universality of TMs. So you can't claim them as "mere". They are intended to represent some meaningful relation expressed as some mathematical operation. They have to perform a function we find useful. Thus they could model a company's payroll, or model the cognitive operations of a brain.
A payroll model is probably pretty ho hum. But a workable brain model?
Yes, the map is not then the territory. As someone pushing semiosis - a modelling relations view of "information processing" - you don't have to explain that to me. It is what I've been saying.
Quoting fishfry
You are convincing me of your utter unfamiliarity with neural networks in practice. Or even in theory.
Quoting fishfry
In fact it is completely custom hardware. It is not a simulation of a neural net on conventional technology. It is a direct hardware implementation of a neural network.
Quoting fishfry
Yes, I've spent 40 years being critical of the over-blown claims of computer science. So I am basically skeptical of the usual talk of getting close to building "a conscious machine". I know enough about the biology of brains to see how far off any computer system still is.
Indeed, I would like it if there was an in principle argument for why no mechanical device could ever simulate the necessary biological processes. It would suit my prejudices. So I am just being honest when I confess that there isn't an absolute argument. The effectiveness of NNs suggests that some level of mind-like technology - as good as cockroaches and ants - may be feasible.
And remember where this started - your claim that abstract thoughts are biochemical processes. You followed that howler by jumping the other way - saying the mind was in no way the product of informational processes.
This second misstep was based on your very narrow conception of information processing - one rooted in TMs.
The reason for the unreasonable effectiveness of TMs is that they are the theoretical limit on semiotic encoding. Semiosis depends on symbols. A TM is the conceptually simplest machine for handling symbol strings.
A DNA strand can code for a pretty vast array of protein molecules, but that’s it really. Human language can code for a vast array of ideas. That's really powerful as we know. But a TM can implement mathematical algorithms. It can articulate any mathematically-constructable pattern. That is a whole other level of semiosis.
So yes. TMs are really basic. They represent pure syntactic potential, stripped of all physical constraints as well as all semantic.
But then we do have to build back the semantics - add the algorithmic structures - to make TM-based technology do actually useful things. Much like DNA has to code for the kind of neural connectivity that can do actually useful things for organisms.
Semiosis recognises the essential continuity here. It sees the ontological difference that codes or syntax makes, the new "unphysical" possibilities they create.
Maybe that's the "physics revolution" you are talking about. I certainly think that it is myself. It explains the information theoretic and thermodynamic turn now happening in fundamental physics I would argue.
I don't believe that intelligibility can extend to fundamentality. So, whatever names we use to denote it: substance, God, the Real, Firstness, the noumenal, the Will, the Apeiron, Buddha Nature and so on, will, with all their associations and connotations, be tools to relate them to our various systematic understandings in the intelligible world, the 'World as Idea' as Schopenhauer calls it.
So, the idea of tychism is really just a dialectical negation of the idea of regularity, stability, concreteness; in short of 'being something'. Spinoza's substance was not thought by him to be "anything", but more like being everything and nothing, inasmuch as to be anything is to be a mode of substance. Hegel similarly said that pure being is close to being pure nothingness.We find apophatic notions of God or Buddha Nature that can be traced back thousands of years. So we can say of Tychism, as Hegel says in another context, that it is the "same old stew reheated".
Quoting apokrisis
Yes, or we can say that reality, 'being something', is an expression of a blind striving that is in itself not yet anything, or, on the other hand, that it is the emanation of an unfathomable, infinite intelligence, or that the 'being something' of temporality is the other face of the 'being everything and nothing' of eternity, and so on. The point of this is that the emergence of concrete somethingness as a process cannot be intelligibly traced back into firstness, because that is where intelligibility ends. We cannot say what is the symmetry of firstness that is broken to produce secondness, unless we impute an intelligence (albeit of an unfathomable order) to firstness, an intelligence of which our intelligence is a temporal reflection. That is the limitation of Schopenhauer's system; it is inexplicable that an ordered Cosmos can be the expression of a blind will. The same goes for any system that thinks firstness as a blind chaos.
My argument is that intelligibility can approach it in the limit - as its own "other". So intelligibility can define the unintelligible as that which it is ultimately not.
And because we know intelligibility to exist, then we know that - whatever else - its unintelligible ground had to contain intelligibility as its potential. So we can actually know something usefully definite about fundamental unintelligibility.
This is apophatic reasoning. But hey, in metaphysics that is unreasonably effective. ;)
Quoting Janus
Exactly. We recover the pre-dialectical through dialectics itself.
Your talk about the fundamental is just dialectics. If we and the Cosmos are an effect, therefore there was a cause. If we and the Cosmos are emergent, then something was the more fundamental.
So the question of creation and being is always going to be dialectical and apophatic. You then need to scout around the history of metaphysics and see who does the job the most rigorously in this regard,
Quoting Janus
I've always said this same old stew has been on the back-burner since the dawn of metaphysical thought. I give full credit to Anaximander with his system of apeiron and apokrisis.
And having checked out many thinkers, Peirce just keeps surprising me with the completeness of his approach. He sorted it out at a fundamental logical level with his triadic model of development. He put the intelligible into the intelligibility.
If you can point out a defect in his analysis, have at it. But telling me others said similar things is not a criticism, is it? My claim is he said it best.
Quoting Janus
So the great unintelligible intelligibility that blindly chose? Does posing an actual contradiction as the origin of being help your case?
Is talk of "emanation" not just hand-waving dressed up in a fancy word?
Quoting Janus
That just isn't a logical argument.
If firstness is where intelligibility ends, then intelligibility is (apophatically) defining it. And for me - given that my worldview is based on the emergence of constraints - apophatic is good. It is fundamental itself.
But you are having to resort to paradox and self-contradiction. You have to talk about intelligences that are unfathomable. You are having to talk about complexity - rational structure - being actually present when there is meant to be only a state of fundamental simplicity.
I just don't get how you can prefer blatantly self-contradicting positions when the alternative is so logical and elegant.
Quoting Janus
Well yes. A blind intelligence is a nonsense. But a blind chaos isn't. It would seem definitional of chaos that it lacks rational structure. So again, I'm just not understanding how you can really believe your own line of argument here.
I get that you are psychologically committed to some notion of a creating God. But so far you are not revealing any hole in a Peircean process philosophy perspective.
Creating gods seem necessary to a certain brand of logic - the one that believes in mechanical or concrete chains of cause and effect. But that is the very logic that leaves out formal and final cause so as to describe the world solely in terms of material and efficient cause. The logic leaves the blank - the absence of formal and final cause - that a creating intelligence then "naturally" has to fill.
So really, in my view, you are just responding to an obvious hole in reductionist cause and effect thinking. It leaves out formal and final cause right from the beginning. So formal and final cause is what you know must be jammed right back in that blank slot.
But Peirce - and all the other systems thinkers and natural philosophers since Anaximander - have a larger dialectic understanding of logic. Formal and final cause have their proper place in the metaphysical system. The blank space left by reductionism is filled by the logic of holism.
And now you don't need some purposeful and transcendent creator. The Comos can spontaneously self-organise out of pure possibility. With Firstness or Apeiron, there is just nothing to prevent that happening, and so it does.
And retrospectively, the outcome will be judged optimal. The Cosmos might start out trying to express every concrete option, but then with all the options in self-competition, the variety of ways of being will be reduced to the outcome that proves the most effective (at being enduring and continuous - or synechic as opposed to tychic in Peirce's jargon).
I don't see how you can deny the simple logic of this application of natural selection to cosmic evolution. Scientific cosmology is now based on this very metaphysics - the Big Bang as a collapse of the universal wavefunction. So it is not as if we lack physical evidence for it. Quantum mechanics tells us classical reality is emergent from a "sum over histories" or path integral. Even a particle gets from A to B by "taking every possible route", and then the actual route is whatever turns out to be the "least action" or energy-optimising path.
So logic tells us the right kind of metaphysical answer. Philosophy of science tells us why reductionist science left the feeling of there being a blank so far as formal and final cause are concerned. And now modern science itself has filled in that blank (or is trying to) by a holistic model in which classical reality emerges from naked potentiality coupled to natural selection.
In the face of all this, you still prefer paradoxical stories about unfathomable intelligences, blind choosing, and "emanations"? Does that really sound like strong metaphysics?
The way this started IIRC is that you accused me of being a dualist and have then proceeded to make a dualist argument for the past several posts.
I actually haven't got much to say at this point and would like to wind this down. I do appreciate that this convo has taken a turn for the civil and interesting. But I'm really argued out on this topic at the moment. I will take your advice and go read up on semiotics. It wouldn't be the first time I've taken a run at that subject but I do admit it eludes me.
But if thought isn't biochemical then what is it? Oh, "informational." But you are using that word in a specific way that's different than what I mean by it.
Quoting apokrisis
Ah yes. Right, we agree on our point of divergence. Well, information processing is a TM. That's the technical definition. You have a more philosophical orientation. You said a post or two back that "information is meaning." I could not disagree more. So we have identified another point of disagreement.
Quoting apokrisis
I must say I actually find that comment interesting. I promise to do my homework on this. The theoretical limit on semiotic encoding. I don't think I understand entirely what it means, but I do find the idea thought provoking. I'll do some thinking and reading.
Quoting apokrisis
But that doesn't prove that minds work that way. Only that NNs have been doing some amazing things. Ever since Deep Blue beat Kasparov I've had to pay attention to weak AI. It's impressive. But I'm sure you know that chess playing algorithms operate very differently than chess playing humans. So my point about birds and airplanes stands IMO.
Quoting apokrisis
No, it's perfectly on target. In the beginning, we tried to program chess algorithms with expert knowledge. (You remember the expert systems movement I'm sure). That got the algorithms to a certain level. But to achieve mastery of the game, the designers gave up trying to teach the machine strategy. They just turned the NN loose and let it train itself. [That's the Alpha Zero approach]. So algorithms play chess very differently than humans do. The bird/airplane analogy is directly on point.
Quoting apokrisis
Well if it's not, then you are making a dualist argument. Because if something is not physical, then what is it? You'll say "informational" and then I'll point out that the mind isn't a TM and you'll say that information processing is not a TM and I'll say you're wrong about that.
Are you making a dualistic argument or not?
Quoting apokrisis
Thank you, I'll read that.
Quoting apokrisis
I said no such thing about neuroscience. I objected to your claiming that neuroscientists think the mind is a computer program. Which is the same exact thing as an "informational process" even though you keep claiming it isn't.
Quoting apokrisis
And physicists keep talking about "gravity" when the question is about falling functions.
If cognitive functions aren't a product of our physical bodies, then you are a dualist. Which is fine, except that we got started when you accused me of being a dualist.
Quoting apokrisis
You're the one who made a sweeping statement about neuroscientists. I pointed out that at best, SOME neuroscientists agree with you and others don't.
Quoting apokrisis
Number-theoretic functions can implement everything else as far as we know. After all a NN is just a set of nodes with numeric weights. An NN is just a set of number-theoretic functions. Turing understood that number-theoretic functions aren't a limitation; but that rather they can be used to compute anything that we think of as computable.
Quoting apokrisis
I'm perfectly willing to grant you that. I even agree with you. But then what kind of computer are they? What kind of computer ISN'T "that kind of computer?" The answer requires a breakthrough in physics and computer science.
Quoting apokrisis
Well summing weights is not an analogy for how NNs work. It's the literal truth of how they work. And by admitting that when it comes to brains, NN's are at best an analogy, you are conceding my point. Brains aren't NNs. You just agreed that they're only analogies to NNs.
Quoting apokrisis
Right. May have. Some future genius is going to have to sort this out because as of today, if something is a computational or informational process, it's essentially a TM. If minds are informational but not TMs, then science has no current understanding of what that would be.
Quoting apokrisis
You are quite the dualist. If something's not physical, what is it?
Quoting apokrisis
How could you have a view either way without a little more computer science to inform your opinion?
Quoting apokrisis
Because we have to go past the Church-Turing thesis to find a mode of computation that's not a TM, and that has the ability to implement mind. And I believe that we'll need new physics to make that breakthrough. I have some ideas along this line but I don't want to toss them out at this moment.
Quoting apokrisis
Mind. Isn't that the "hard problem" as they say? We're trying to figure out how a body can have a mind. It's real puzzler.
Quoting apokrisis
If brains implement minds and brains aren't TMs then what exactly are minds? Isn't that the question? I don't have the answer but nobody else does either. Not even the neuroscientists, who you seem to think have already settled the matter.
Quoting apokrisis
Private between me and every computer scientist and information theorist in the world. Information is a bitstring that can be cranked out by a TM. Randomness is a bitstring that can't. Kolmogorov. He and I inhabit the same private world, although he's a hell of a lot smarter. And deader.
Quoting apokrisis
I hardly regard math as part of the real world. it's literally the opposite of the real world. And philosophy has even less of a claim on being real. Surely you know this.
Quoting apokrisis
If the next bit's not a surprise, it's because there's a TM behind the curtain. Which reminds me a little of the intuitionists, who think there's some kind of active agent creating "free choice sequences." I tried to understand that idea once and gave up. Is that what you are referring to here?
Quoting apokrisis
Of course. Pi is a computable number. We only need to write a program to implement any of the many known closed-form expressions to calculate its digits. You know you're drifting into agreeing with me again.
Quoting apokrisis
LOL. By which you mean, "You have an opinion that differs from mine, and I haven't got much of an argument."
Quoting apokrisis
Um ... that's a little murky.
Quoting apokrisis
Yes ok.
Quoting apokrisis
Murky. Bitstrings generated by TMs we call information; and those not generated by TMs, we call random. Quantifying semantics is a mystifying phrase to me. I'm not saying it's meaningless or wrong, just that I don't know how to quantify semantics or what that would mean.
Quoting apokrisis
The habits of the human minds that assign meaning to the symbols. And even that is a function of language. A cat is a gato is a chat. Meaningless strings of symbols, assigned meanings by humans.
Quoting apokrisis
Actions of humans.
Quoting apokrisis
Causally? No. The link seems arbitrary. As George Carlin asked, why do we park in the driveway and drive on the parkway?
The link between words and their meaning is anything BUT causal. It's arbitrary. Isn't it?
Quoting apokrisis
Murky. You were making the claim that there's a causal connection between the arbitrary string of symbols "gato" and a furry four-legged domesticated carnivorous mammal with soft fur, a short snout, and retractile claws. You're wrong about there being a causal relation. But this quoted sentence seems to come out of nowhere and mean nothing.
Quoting apokrisis
I promise to do so. I have a lot of research and thinking to do as a result of this thread.
Quoting apokrisis
I'm shocked. You have an argument? Sorry couldn't resist. You keep misrepresenting my argument too. But I think we're really repeating ourselves at this point. I really have to stop responding.
Quoting apokrisis
Well then you have backed off your claims and essentially agreed with me. I am in full agreement that NNs have done some amazing things in constrained domains. That does NOT mean the brain is an NN. Please don't make me say birds/airplanes again.
Quoting apokrisis
LOL. That's like me saying you're confused about Peirce. It would push your buttons only for the sake of doing so.
I'm not confused about algorithms. I think very clearly about algorithms. That's why I know that algorithms are dumb as rocks. And that's an insult to rocks.
Quoting apokrisis
Oh My God. No. TMs are not semantics. I'm sure you didn't mean to write that. I hope you didn't. Algorithms are syntax. Their semantics comes from humans. Please tell me you understand this.
Quoting apokrisis
Not in my world. Not in anybody's world. Computations are bit flipping. Pure syntax. The meaning is entirely supplied by humans. Remember the 80s movie War Games? The computer doesn't know the difference between playing a game and blowing up the world. To the bit-flipping circuits there is no difference.
Quoting apokrisis
The intention comes from the humans. They are "intended." But the intention appears nowhere in the code.
Quoting apokrisis
It's only flipping bits. The meaning is in the intention of the human programmers. Play a war game or blow up the world. The computer just flips the bits, it doesn't know the meaning of anything. I really hope you are not confused on this point. I have no idea why you are trying to say something that you must know to be false.
Quoting apokrisis
To be picky, a UTM is a universal algorithm runner. A specific TM only runs one algo. Minor quibble.
Quoting apokrisis
we have in mind
Yes. You agree with me totally. We supply the meaning. The meaning comes from our minds. There is no meaning and no mind in an algorithm or in a computation.
Quoting apokrisis
Could be. Yes. Agreed.
Quoting apokrisis
Right. Why do you so strenuously agree with me?
Quoting apokrisis
No. I'm not confused. An algorithm is a description of a computation. A computation is a physical implementation of an algorithm. Information consists of bistrings cranked out by programs.
Quoting apokrisis
Well algos only model a very limited subset of brain activity. And even if you had (in the future) an algo that could simulate the physiological functioning of a brain, it still would not implement a mind. Just as a computer simulation of gravity does not attract nearby bowling balls.
Quoting apokrisis
Just a matter of time an neuroscience. But you are conflating brain with mind. I fully agree that one could in principle someday have a computer program that performs a reasonable simulation of a brain. I deny that such a simulation would have a mind. I agree that one could disagree with what I just said.
Quoting apokrisis
Well we have been in agreement for quite some time I think.
Quoting apokrisis
In other words you have no argument so you toss out a little gratuitous snark. It's ok, it's the best you can do.
Quoting apokrisis
It's still conventional computing. Still a physical implementation of a TM. If your friend says he has a computer whose code and/or data can't possibly be backed up, he's lying. What happens when there's a power failure? He types it all in again? I'm aware that there is custom hardware to implement NNs. But it's still based on the conventional model of computation.
Quoting apokrisis
As have I. I read Dreyfus's What Computers Can't Do many years ago and it's influenced my thinking about all this.
So why are you so hell-bent on convincing me that I'm wrong when I am in full agreement with everything you say except for your tragic misunderstanding of computer science?
Quoting apokrisis
Which is exactly what I've been saying all along.
Quoting apokrisis
I see no disagreement between us here.
Quoting apokrisis
It started with you calling me a dualist and then pushing dualistic ideas yourself.
Quoting apokrisis
It's not, if by informational processes we take the standard contemporary definition in computer science.
Quoting apokrisis
If you have a better idea, and not just handwavy misunderstanding, I would like to hear it. If you have figured out how to break the Church-Turing barrier it will make the news because it would refute an 80 year old orthodoxy.
Quoting apokrisis
Actually there are much simpler models such as finite automata. But your statement about the limit of semiotic encoding, that's something that I have to go find out more about. It sounds interesting.
Quoting apokrisis
I could write a program that would simulate everything we know about DNA (assuming I took the trouble to learn about DNA). In fact it's the informational basis of life that is the strongest evidence for the point you're trying to make. The next leap is to see how perhaps mind might actually be informational yet not a TM. Or perhaps we're all just programs. It's possible. I don't think so and I don't want to think so, but maybe we're all characters in a Philip K. Dick novel who wake up to discover we're just robots.
So yes. TMs are
Quoting apokrisis
Yeah yeah. We've both run out of stuff to say.
But then we do have to build b
Quoting apokrisis
DNA is the programming of life. But is it the programming of mind? I don't think that discovery's been made yet.
Quoting apokrisis
Unphysical. So we're dualists again?
Quoting apokrisis
The revolution IMO is to break the Church-Turing barrier. We have to find a mode of computation that lets us say that mind is a computation but not a TM. It's my belief we'll need new physics for this.
There's a difference between substance dualism and my dialectical or semiotic approach.
Quoting fishfry
If that were true, computation becomes a physical impossibility. The technical definition requires the physical manipulation of an infinite tape. Those are quite hard to come by in the real world.
So yes, we can pretend. We can build pseudo-TMs that paper over this embarrassing fact with virtual machine architectures. All you have to worry about now is what you do when you exhaust 64-bit memory addressing, and then 128-bit, etc.
Again, you speak so confidently about neuroscience and computer science. But yet you seem to confuse theoretical constructs with real world practicalities. As was my point, TMs define an ideal limit. And thus a literal TM is also physically unrealisable.
Quoting fishfry
I can't take you seriously when you make such weak arguments. Of course it is evidence that we are getting at something central to the functional design of the brain. Just as being able to mechanise bird flight would have been evidence we were capturing the essence of the way birds fly.
And just as we instead built fixed-wing planes as the unsubtle brute force alternative, so computers are the familiar clunky von Neumann architectures they have been since computing got properly started. There is nothing biologically-inspired about the design. Yet they do the job - given our limited practical purposes of just getting around or automating various tasks.
Quoting fishfry
Yeah. I do remember. And NNs were pushed before that. In the beginning, more naturalistic architectures were being suggested. Check out cybernetics. But then symbolic processing became the 1980s fad. Lisp machines and all that. I happened to edit a computer journal at the time expert systems were getting hot and hyped. The history of all this is familiar.
Quoting fishfry
Stop misrepresenting me. I didn't say neuroscientists think the mind is a programme. And information processing is more broadly defined than by universal Turing computation. Before digital there was analog computation or a start. Learn your history and stop making a fool of yourself.
Quoting fishfry
It's not an admission. It's my point. NNs are successful models of the brain's essential functional architecture.
You talked vaguely of "biochemical processes". Well science prefers to talk precisely. And it seeks to understand the basic trick of the brain in terms of some replicable informational architecture.
But hey, I've lost interest. If all there is to do here is to keep correcting your misrepresentation of my arguments, that is really a waste of time.
You mean an unbounded tape. Your ignorance is showing. Again.
Quoting apokrisis
No amount of erudition can compensate for your bad manners. And as in our previous convo a few months ago, the more wrong you are on the facts [which several people noted at the time] the more condescending you act. Transparent and shallow, you fool no one but yourself.
I don't disagree with anything here. We apprehend and comprehend the idea of finite, temporal being, and understand that it cannot be its own ground, which leads us to the idea of in-finite, eternal being. So, my question is, just as with finite temporal being we extrapolate to in-finite, eternal being; then why not from finite, temporal creativity to in-finite, eternal creativity, from finite, temporal intelligence to in-finite eternal intelligence, from finite, temporal order to in-finite, eternal order, and so on?
In any case whatever we think about that in-finite, eternal being is always going to be tendentious, always going to reflect our own prejudices; if we are moved to take any position as regards its 'attributes', it can only be on the basis of imagination mediated by logic; it can never be an empirical matter.
The extrapolation has to reverse a dichotomistic separation. It has to unbreak a symmetry breaking to recover the original symmetry. So there is a particular logical model to be followed.
So for instance, if existence depends on the actualised contrast between flux and stasis, or chance and necesssity, or discrete and continuous, or matter and form, etc, etc, then that definitely present distinction is what has to be folded back into itself as we wind back the clock to any vague and undivided initial conditions.
It is the unity of opposites argument. Dialectics. And I don't see that you are posing your own account this way.
The infinite is opposed to the infinitesimal. They represent the limits on the dichotomy represented by the notions of the ultimately continuous and the ultimately discrete. So a symmetric initial conditions would fold this distinction back into itself. It would be a state that is neither infinite nor infinitesimal. Neither continuous nor discrete.
That of course sounds rather mystical. But it actually maps pretty well to the Planck scale which was the start of the Big Bang. The geometric extent of spacetime was infinitesimal - as little as it could possibly be. While the energetic content of spacetime was infinite - as hot or dense as it could be.
So there is a logical formula to follow here.
You, on the other hand, are imagining a linear extrapolation. You start with some limited amount of something and multiply it until it grows to be unbounded. Time, creativity, intelligence, order and being are all finite and definite properties, so why can't they be - individually - infinite?
So nothing is being folded back into itself to heal a symmetry-breaking. There is no dissolving of the crisply divided to arrive back at a shared primal origin. The metaphysical operation you have in mind is instead turning a limited substance into an unbounded substance.
Instead of dissolving hylomorphic being by folding form and matter back into themselves via a loss of all distinctions, as in the notion of an Apeiron which is just pure fluctuation, you are accepting the substantial state and extending it without limit. It loses its located particularity by being rendered absolutely general rather than by being dissolved back to a vagueness.
The Peircean model is firstness => secondness => thirdness. Or vagueness => particularity => generality.
So sure, you can make absolute generality your initial conditions rather than your final outcome. But that then rules out a developmental logic.
You can see this tension playing out in theistic attempts to imagine divine immanence. Is God there at the beginning or realised at the end? Is God the creating intelligence who decided to construct a Cosmos for some reason, or is the Cosmos, through its evolution, the eventual realisation of Godhood?
Peirce's metaphysics argues that the development of the Comos represents the universal growth of reasonableness. The beginning was an unintelligible chaos - meaningless tychism. But that couldn't help but develop patterns and order. Habits emerged. The Cosmos started to self-organise and become intelligible. So the Cosmos is on a journey towards maximal "reasonableness". To the degree you want to read divinity into the story, the "designing intelligence" is simply the semiotic machinery - the fact of habit-taking - by which chaos can become completely ordered in a general or global fashion.
So this would be divine creation or divine intelligence of the most limited kind. Especially right back at the beginning. And even at the end, it only manifests as some general state of order. It is not intelligence as we mean it - a mind cracking problems for self-interested reasons. It is simply a mechanism - semiosis-driven self-organisation - extrapolated to the most global possible scale of being.
Again, modern science confirms this particular metaphysics. The Big Bang is self-organising its way to its Heat Death. The Planck scale symmetry breaking will become eventually as broken apart as it can physically be. It will arrive at the stasis of an anti-de Sitter void, a fully thermalised and unchanging dead universe.
Creativity and intelligence and mindfulness as we mean it are just passing negentropic eddies in this general flow towards a maximum entropy condition. We are not what creation is about. Even if we exist only by contributing to that entropification project.
So to extrapolate from us is metaphysically unjustified. At least if we are following a metaphysics that is based on the logic of dialectics.
And given that metaphysical dialectics proves itself to work, we should be brave enough to follow it all the way to talk about the beginning and end of substantial existence itself.
It is fine that you make your extrapolation argument that starts with the particular and abstracts to the general. But an unbounded amount of some stuff - like time, intelligence, creativity, whatever - is not actually a shedding or dissolving of boundaries. It is only a generalisation that leaves you with an unlimited quantity of that very stuff. The stuff is still bounded, still substantial and definite, even if you are imagining it to be actualised in some infinite quantity.
That's the problem. Stuff is hylomorphic. Definiteness is defined by the existence of a dichotomy - the unity of a complementary pair of bounds or limits. Your extrapolation only multiplies the quantities. It cannot dissolve the actual qualities in question. And so talking about an infinite amount of something fundamental solves nothing, just multiplies your causal difficulties.
A finite amount of intelligence or creativity is easier to explain than an infinite amount. And at least a finite amount, if multiplied enough, could become an infinite amount - using our metaphysical maths.
But to solve the problem of creation, the question of how existence could bootstrap into being, you need to be able to dissolve substantiality itself. You must undo the very notion of a quality.
And dichotomies - in defining a reciprocal or inverse relation - can do that. If you multiply x/1 by 1/x, you get 1. You can unmake your perfect asymmetry and recover a perfect symmetry. So now the multiplication goes in the right direction. The particular become not the general but the vague. You can no longer tell one pole of being from the other. Their particular qualities have been merged - mutually annihilated - to become again a featureless one-ness of unity of opposites.
So we have here two views that can be defined as mathematical operations. The metaphysical claims can be made highly precise.
The question then is which one is actually doing the trick? And which metaphysicians have talked about the opposed alternatives the most clearly?
I don't have time to respond in detail; but I will just point out that by 'in-finite' I don't mean to refer to "an infinite amount"; all amounts are finite. Likewise 'eternal' does not mean 'for an infinite amount of time" but 'timeless'; outside of the context of time altogether. The "virtual quantum soup" must be thought as both in-finite and eternal (not spatio-temporal, in other words) according to these definitions. It is not non-being in the sense of absolute nothingness (which is an absurd notion) but infinite, eternal being. If it "contains the potential" for intelligence and intelligibility, then it would seem to make more sense to think of it as in-finitely and eternally intelligent, than to think of it as brutely blind.
Yes. You sense the difficulty and try to avoid it.
I seek to make the difficulty plain and so force a definite choice.
Quoting Janus
But here you have to offer the dichotomy on which your notion of intelligence, or intelligibility, or creativity, etc, is based.
You have to show that you are un-breaking a breaking rather than extrapolating a quantity to arrive at an "unbounded" quality.
My view of the development of human intelligence and creativity is scientific. It has that tested evidential support. And so I don't think of these named qualities as being in anyway physically fundamental or general. There is nothing more particular and emergent in the known Universe than the complexity of a living human nervous system.
So the dichotomy, the formal contrast, is between complexity and simplicity, between negentropy and entropy, between organismic level self-interest and purpose and physical level disinterest and blind tendency.
Peirce then connects the simple and the complex in psychological or phenomenological language. Intelligence (or any evolutionary/adaptive process) needs to combine selection pressure and spontaneously arising variety. Hence we arrive at the story where firstness equates to absolute blind spontaneity or tychism, and thirdness equates to absolute firm habit, the continuity of evolved constraints or synechism.
So now nature can be "intelligent" in that it has this evolutionary logic, this intelligible structure. A fruitful marriage of chance and necessity, freedom and constraint.
Human-style intelligence and self-centred purposefulness falls out of the picture. It is a general possibility taken to a particular extremity. To talk of a still "higher" creating intelligence has to be a continuation of that particularisation. A super-mind would have to inhabit a super-body as well.
Now we can imagine such a next step. The idea of artificial intelligence and the Singularity is one such extrapolation. Humanity could get downloaded to a technology that spreads itself across inter-galactic space.
A nice conceit. But it does correctly extrapolate whatever it is that we could mean by human intelligence and creativity as qualities that might increase in general quantity. If life and mind is negentropy that depends on accelerating the Universe's entropification, then spreading ourselves across the Universe to tap its physical resources at every possible location is what natural philosophy would predict.
But anyway, this is the issue. You have to choose whether you are magnifying a quality or dissolving a quality. And how can you head back to the origins of a quality by simply increasing the amount of it?
You claim you are not increasing the amount in trying to generalise the quality. But really, you are. You are imagining a little bit of local stuff spreading to take over everything. And that is simply shifting reality in the direction of one pole of some dichotomy. You are arguing that eventually - go far enough - and you lose sight of the other pole.
So take embodied human intelligence and creativity. You want to lose the necessity of the body and imagine the mind spread generally.
The Peircean view is pragmatic - mind arises as a way to regulate material physics, accelerate entropic flows. Mind makes no sense, it can't exist, unless it has that physical context.
So you are imagining a nonsense - a mind without that "other" which is the source, the cause, of its being.
I can see how tempting this move is. We are so used to thinking in terms of dualism. It is simply believed that mind and world are already separate, so both are free to grow in-finitely in their own realms.
But that is a dualistic metaphysics. And it doesn't in the end work. We know that. Hence even theists do try to find a more organic or immanently self-organising story occasionally. And Peirce spells out the logic of that.
In-finite mind is not "spread" anymore than infinite being is. You need to free your thinking from it's customary presuppositions to get this.
Quoting apokrisis
No, I'm not proposing any kind of dualism; that it might seem so is again due to your own prejudice. How can you tell, beyond its failure to gell with your own particular set of presuppositions, that a metaphysics is not working?
Or you could explain what you are thinking as the alternative.
What even is in-finite being? You could be agreeing that it is the Apeiron - a potential not yet limited and thus not yet actualised. Why not explain in your own words to make it clear.
And what then is in-finite mind? If we follow the same formula, it is a potential that is not yet intelligibly structured and so not yet "intelligent".
Or maybe you are talking about sentience, or qualia, or soul, or something similarly substantive - a universal simple. So now you are either defending dualism or idealism. That becomes un-Peircean as Peirce is all about the psychological structure, the growth of universal reasonableness.
The Peircean claim is clearly argued in terms of an actual logical mechanics. The process is laid out in plain view.
Can you do something similar for "in-finite mind" if it is really universal sentience you are talking about, or universal will?
You talked about intelligence and creativity. That fits with a basically physicalist approach like Peirce takes. It avoids just presuming a dualism or idealism as the ontology.
But if you really meant in-finite sentience or in-finite will, that is what you need to defend.
Quoting Janus
Just because I tell you plainly what my position is, and then ask you to be plain about yours, doesn't mean I can't get past my beliefs.
It means I have arrived at my beliefs through a contest of the alternative views. And one of the fundamentals of epistemology is that offering up theories that are "not even wrong" is worse than a concrete theory that just is wrong.
So again, what I am hoping is that you will put forward a sharper account of the world-creating mechanism you have in mind as an alternative here.
If you are not talking dualistically, then what is this "mind" of which you speak? Is it synonymous with being, and so monistic in the idealist sense, or what?
If you leave me guessing, you can't really complain if I fill in your side of the debate too.
The problem is that an infinite pure potential that is not actual makes absolutely no sense. Remember we are both doing no more than trying to speak suggestively, apophatically, here about 'something' about which really nothing can be sensibly said.
That doesn't help me understand what you could mean by in-finite mind here.
And I already said that - apophatically - the unbounded initial potential would be defined in terms of it being "not actual".
It's the same way we talk about "nothingness" - the absolute absence of things. But with nothingness of course, there isn't then a potential. Potentiality is what has been absolutely suppressed. So the difference with an Apeiron, Firstness or Vagueness is that we know it must have had a potentiality that was un-actualised. We know there is that actuality. So if we read it correctly - in terms of symmetry-breaking - that justifies our saying something concrete about that which is not the least concrete.
Again, I believe I have spelt out a metaphysics with an actual logical machinery. It is even mathematical in being framed in terms of reciprocal or inverse relations. It is certainly scientifically inspired in being a tale based on fundamental symmetry breaking.
So that is why I ask you to offer something as well developed if you want to argue for "in-finite mind". It seems fair enough to me.
I'll try to address this more later when I have time. For now I'll just say that the idea of an infinite potential that is not actual makes no sense to me. I mean what is the opposite of an actual potential? An imaginary potential, or a conceptualized potential?
Saying a potential was actual is the after the fact view. So it says something was possible rather than impossible.
An infinite potential would then make anything and everything possible. At least at the "beginning". What is then now the actually possible vs the actually impossible is whatever actually happened and whatever actually didn't.
The reason some potential things would be impossible would be because constraints emerged to limit their being. Their existence was suppressed. And thus actuality gets defined by whatever it is that constraints can't suppress. Actuality is an expression of what can freely happen.
So you are wanting to make the initial potential some kind of concrete actual. You want it to be already substantial - limited by form so that it has definite being ... ahead of there in fact being any definite being.
But that is the mental hurdle you need to move past here. And I agree it is really difficult.
That's the problem; I find it impossible to think that absolutely anything at all would have been possible even 'at', or 'prior to' the 'beginning'.
A universe of cartoon characters? A universe which is just a giant hamburger? A universe consisting of fairy floss? A universe where the inhabitants are heavier than the planets they inhabit? An infinitely complex and changing world which nonetheless consisted in absolute thermodynamic equilibrium? Or could any world such as our present one simply pop into existence 'fully formed' and without a history? I mean imagination's the limit; I believe I could think of potentially an infinite number of scenarios which would simply seem, not logically, but inherently physically, impossible, period.
So it seems impossible for me to imagine that there would not be an actual lawfulness inherent in the primordial indeterminate potential, that always already limits what could possibly come to exist. Of course, it's all speculation; but what tools do we have to work with other than imagination, intuition and the logic of what seems possible? I mean, we are endeavoring to determine what we should think about extremely abstruse matters, after all. And I do think it should, and probably inevitably will, remain ultimately an individual matter. We are not constrained by what the "community of enquirers" will ultimately come to think, because we cannot have any idea what that will be, or whether there will ever in fact be any overriding consensus.
Have you already abandoned apophatic reasoning?
We start with where we are. We accept that there is a creation issue because we have the clear evidence that our existence has developed. Cosmology and fundamental physics then let us wind back our story all the way down to a quantum-level scale where radical indeterminism is known to set in - the Big Bang scale of about 10^-33 cm, 10^-44 secs and 10^32 degrees.
So it is from that set of physical conditions that the metaphysics continues to extrapolate.
Yes, imagination is needed. But it is now extremely constrained by the facts we are sure of. So all your suggested worlds, chosen because they are silly and contradictory, are already ruled out - unless apophatic reasoning finds some way they are a logical consequence of the "whatever" which would be the kind of potential which also produces such a highly constrained Cosmos such as the one that works to produce us.
So the question about other possible worlds would be about physical basics such as the number of dimensions, the strengths of constants, the number of different emergent forces. And none of your imagined universes suggested something different about those.
Quoting Janus
Well it helps to start by getting down to a starting point based on what we know. So it we are looking for what lies immediately before the Big Bang, we know that we are talking about some kind of quantum mechanics that lacks the kind of dimensionality which gives quantum fluctuations a strength and a direction in "our" universe.
And there is a ton of speculative physics on the issue if you want concrete proposals. There are models like the many loop quantum gravity approaches that seek to show how our 4D spacetime world could arise emergently from naked fluctuation. There are thousands of papers on the issue. There are computer simulations of self-organising spacetime metrics. People are trying to bring the right mathematics to bear on the question.
Quoting Janus
Again, the scientific community is on to it. It's not about personal belief. It is going to be about whatever mathematical-strength model shows our particular dimensional set-up was always necessarily emergent from whatever an utter quantum indeterminacy can be understood to be.
There's a well defined approach and goal here. It's actually a pretty interesting story unfolding before our eyes if you check out the science.
A number can be many things. It can be careful symbol that we hold on memory. It can be a multiplicity that we hold in memory of something we observed or possibly created in our memory. They all exist as memory but not necessarily shared in other memories. I prefer thinking in terms of memory that is private and memory that is shared since everything ultimately is memory of our mind of some sort.
Quoting apokrisis
So, the ways things are in our world that has emerged constrains how we can think about the the primordial, in-finite, eternal, virtual 'anything is possible'? We know what it is not and it is not anything we know?
Youo recommend checking out the current developments in science and I do find the idea of systems science and semiotics based metaphysics intriguing, but I am weak on math, and short on time, so I not confident I am able to adequately judge the validity and soundness of the kind of work you are recommending, since the validity and soundness is founded on the math. Also, I have always been drawn to the free flights of imaginative and intuitive thought, simply because I find I can do it, and I have a strong sense of the numinous which I am disinclined to give up on account of a belief that to do so would be to impoverish my life.
You have often accused me of being committed to a belief in a creator god, but that is not really accurate. I don't have any clearly defined belief about the ultimate nature of reality, and any religious feelings I might have had have not been strong enough and/or of the appropriate kind to have made me a church-goer or adherent of any particular religion.
I asked you before what would be a universal presuppositon-less criteria for judging whether a metaphysics "works". Is such a thing possible? I tend to think that the metaphysical ideas that are the most exciting and inspiring and able to elaborated into the most comprehensive systems are the ones that work best, since metaphysical ideas cannot be judged in terms of practicality, or reliably inter-subjectively corroborated.
At the moment I'm exploring Whitehead's works to try to gain an understanding of the overall movement of his thought. Some parts of The Concept of Nature (which I'm currently engaging with) are couched in mathematics, and I have a terrible time trying to prevent a loss of enthusiasm and interest when confronted with mathematics.
That's fine. You seem interested and serious. I am just arguing for a particular point of view which represents a logical metaphysical methodology.
Quoting Janus
Any reasoned position must start from suppositions. How else could it work? The alternative would be perhaps some claim about "direct perception" - personal revelation - in regard to the truth of the Cosmos.
So yes, there must be some well-chosen suppositions to get the metaphysical game going. As I've said, the first is that something exists. The second is that something developed. The third is that development must be dialectical or dichotomous - a separating out or symmetry-breaking which speaks to a prior unbroken potentiality. The fourth is then that we must be dealing apophatically with some kind of "perfect potential" as the initial conditions.
Whether this metaphysics "works" as a whole then depends on how other alternatives stack up against it. It could either be challenged at some particular step (perhaps the Big Bang never happened, there was never a developmental story), or it could be challenged as a whole - as transcendent theism would likely do.
Quoting Janus
I find Whitehead dire. Personally I would waste no time there.
The beauty of mathematics lies in its Platonic structures. It is the fact that fundamental abstract patterns can be the bones that underlie the flesh and blood materiality of the world.
A better read than Whitehead would be something like Ian Stewart's Why Beauty Is Truth: A History of Symmetry.
Your suppositions do make sense to and I dare say they would be essential to almost any conceivable consistent metaphysics.
I know what you mean about Whitehead, but I have been looking into him on and off over a period of perhaps 20 years and am gradually coming to comprehend the range and cohesion of his whole sweep of thought. So, I have found quite a deal of interest in his work.
Thanks for the Ian Stewart citation; I'll check it out. I have Nature's Numbers on my shelves. I've been meaning to read it for years.
It's a philosophical curiosity that there is no definition of number in mathematics. In other words if you major in math, get a Ph.D. spend a career as a professional mathematician, you will never encounter a book or a paper that says, "A number is such and so."
A number is pretty much anything that's number-like, and the concept is historically contingent. First numbers were only the whole numbers and the ratios. Pythagoras discovered irrational numbers. Not till the middle ages did we start regarding zero and negative numbers as numbers. Not till Cantor did we regard transfinite quantities as numbers. There are all kinds of other mathematical objects we call numbers such as the p-adics, the dual and perplex numbers, nonstandard integers and reals, and so forth.
Yet there is not one single definition of number. It's an amorphous concept. Mathematicians "know one when they see one." I don't know if this has caught the attention of philosophers. But there is no definition of number.
Ok I'll play. Four questions.
* What is quantity?
* The imaginary unit i with i^2 = -1 ... what quantity does it represent?
* Do you regard i as a number?
* Does i exist?
Probably worth mentioning that category theory and structuralism have moved past this good old set theoretic view....
So if it looks like algebra - it can be added, subtracted, multiplied and (perhaps) divided - then its all numbers.
Really? What is number theory about? What is Principia Mathematica about?
Sorry but you are bullshitting to an extraordinary level.
Quoting fishfry
You think numbers are defined in terms of quantity?
i represents the square root of -1.
i is a number.
i exists.
Simples.
Saying that a number is anything that's number-like is a circular definition. No better than the poster above who said that a quantity is anything that's quantitative. You are defining a thing in terms of itself. It's not a definition.
Even worse, being "number-like" is neither necessary nor sufficient for something to be considered a number! Some examples:
* Matrices can be added, subtracted, multiplied, and sometimes divided. In fact the set of nxn matrices for fixed n forms a ring, an important algebraic structure. But matrices are not regarded as numbers.
* The set of permutations on a given finite set forms a group. An example would be the six permutations of the set {a,b,c}. Geometrically this is the group of symetries of an equalateral triangle.
Permutations can be combined via the operations of composition: you do one permutation followed by another. With this operation, the set of permutations is a group. You can multiply and divide. But nobody ever calls a permutation a number.
* If we have a pair or an arbitrary collection of algebraic objects such as groups, rings, fields, vector spaces, or modules, we can form their direct sum and their direct product; and in the case of modules, their tensor product. These sums and products obey various algebraic identities and are considered part of the study of algebra.
But nobody ever calls groups, rings, fields, etc. "numbers." They're algebraic objects but they are not numbers.
These examples show that there are things that are "number-like" that are not numbers; and things that are not numbers that can nevertheless be added and multiplied.
I make two assertions:
1) In math, there is no general definition of number. Of course there are perfectly clear definitions of certain classes or types of number: integers, reals, quaternions, p-adics, transfinite ordinals, etc. But there is no general definition that says, "A number is such-and-so."
I was surprised at the, let's say, passion of some of the responses to this tame and factual assertion. Many interesting points were raised. I'll try to respond in detail to each post in the next couple of days.
2) It's surprisingly tricky to give a good definition of number. I hope my examples bear that out. But if they don't, I have a lot more examples!
No it's not. It's defining something in terms of its relational qualities rather than in terms of its supposed essences.
The whole point is that "number" is elusive as an "abstract object" because that is wrongly to seek some constant essential thing that is more primary than the relations that ensue. So the right way to look at it is to switch to a contextual, constraints-based, metaphysics where objects are defined in terms of structures of relations.
But if you want to complain, write a letter to the Department of Category Theory. Let them know it is from the Department of Set Theory. That will help them make a speedy decision just where to "file" your complaint. ;)
Quoting fishfry
Not even complex ones?
Quoting fishfry
Or maybe you are wrong?
Quoting fishfry
It seems curious that you can both claim numbers don't have a good definition and then so easily rule lots of things in or out as numbers.
I wonder what criteria you use?
Quoting tom
Ok good questions. First to clarify what I'm talking about, my totally humble thesis is:
There is no general definition of number anywhere in mathematics. Of course there are perfectly clear definitions of particular types of numbers. Integers, reals, quaternions, p-adics, transfinite ordinals, and so on. But nowhere in mathematical literature will you find anyone who ever says: "A number is defined as such and so."
I regard this as a philosophical curiosity, worthy of discussion. What surprises me is people claiming that my thesis is factually false.
Now to falsify my thesis, all anyone needs to do is post a reference to a math textbook or published paper where someone defines what a number is, in a way that other mathematicians have adopted (ie not specialized to that one paper).
Post that reference and my thesis stands refuted.
All the verbiage in the world that is NOT such a specific reference does NOT falsify my thesis.
I trust this is perfectly clear to fairminded philosophers. I am not making any metaphysical statement about numbers. I'm saying that in the mathematical literature, there is no general definition of number.
To falsify a universal statement it is both necessary and sufficient for you to supply a single counterexample.
And if you DON'T, my thesis stands till you do.
* Now you did reference two interesting topics. First, number theory at the elementary level is about the study of the integers, and sometimes just the positive integers. These two classes of numbers have perfectly good definitions. At higher levels. algebraic number theory is the study of the algebraic integers. Analytic number theory is the use of the techniques of analysis: limits, calculus, etc., to study integers or algebraic integers.
In all of number theory, there is no definition of number.
* Now the Principia is cool. It's one of my favorite things. In the Principia, Newton describes the fundamentals of calculus; and uses his techniques to prove that the planets and the apples on the trees obey a universal law of gravitation that can be described by a simple equation.
Now that's cool as hell. He was one smart cookie that Isaac.
But I can tell you for a fact that in all of the Principia there is no definition of what a number is. Why would Newton concern himself with a matter as trivial as that? He was laying out how the universe works, not quibbling about definitions.
As an aside, Newton had worked out calculus using a symbolic formulation; but he wrote the Principia using the geometry of the ancients. He knew that he was proposing radical ideas, and he wanted to use familiar mathematics. He didn't let his new notation get in the way of the acceptance of his physical ideas.
Another reason Newton deliberately obfuscated the Principia was, in his own words, "... to avoid being baited by little Smatterers in Mathematicks."
I can totally relate!
Quoting tom
I wish you could explain this to me. I stated that in all of math there's no official definition of a number.
Why does this push your buttons? Why would you even doubt me? I've done the research. It's not hard to verify. I think there might be a Stackexchange thread about it.
Why do you think I'm bullshitting about this? And really, if you think of it, why WOULD I bullshit about something like this? What's in it for me?
Quoting tom
No, you didn't read far enough back in the thread. Someone proposed the definition, "A number represents a quantity." I proposed the counterexample of the imaginary unit i, which does not denote a quantity in any sensible interpretation of the word. And I also asked that poster to define quantity for me. So that I would know for myself whether i could be somehow interpreted as denoting a quantity.
So I was asking the question as part of my challenge to their proposed definition of a number.
You understood me to be claiming that a number represents a quantity, but of course that is NOT a belief I hold. Complex numbers, p-adics, transfinite ordinals. I would never say a number represents a quantity.
Quoting tom
Agreed, though it's more precise to say that i represents "a" square root of -1, because -i is another one, and i and -i can't be distinguished. There's no sense of positive or negative in the complex numbers.
Quoting tom
Yes of course. I believe in the complex numbers.
Quoting tom
Yes it does. It not only has existence within formal mathematics; it occurs everywhere in the real world. If you want to do quantum field theory or just turn left at the corner, you are using the number i.
Quoting tom
Absolutely. Since we share the same mathematical ontology, I wonder what I said to upset you.
At least Frege, Russell, and Whitehead defined what a number is. There are probably several others.
Hang on, there's even a Wikipedia page:
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
Quoting fishfry
Follow the Wikipedia links, do some Googling, you are refuted.
Yes that's an interesting point. Philosophers and logicians have struggled to define what a number is. Mathematicians don't really care that they haven't got a precise definition. Mathematicians expend zero energy going down that rabbit hole. It's not mathematically productive.
There is an advantage to this approach. Mathematicians are not constrained by a definition of number, which allows them to discover new types of numbers all the time.
Quoting tom
I have repeated the same thing several times, yet you are still misunderstanding what I'm saying.
There are perfectly clear definitions of specific types of numbers such as naturals, integers, quaternions, etc.
And in fact each foundational approach has its own definition. In set theory the natural numbers are defined as the von Neumann ordinals. In category theory there's a natural number object, as @apokrisis mentioned earlier.
The page you linked to is the von Neumann definition of the finite ordinals, which has the nice advantage that it can be easily extended to transfinite ordinals.
In category theory there's a concept called a natural number object. This defines the natural numbers structurally, as @apokrisis noted earlier. The benefit of the latter approach is that it avoids the so-called "junk theorems" of the von Neumann approach. For example in standard set theory, 2 ? 3 is a valid theorem. No sensible person would claim it means anything. The categorical approach gets rid of that type of problem.
My thesis is entirely agnostic of foundational approach, a point @apokrisis does not sufficiently appreciate. There is no general definition of number in set theory, category theory, homotopy type theory, Martin-Löf type theory, intuitionist type theory, any of the various constructivist ideas, or any other foundational approach. There are many foundational approaches these days. My statement applies to all of them.
There is no general definition in math that tells us what a number is. There are plenty of definitions of specific types of numbers. There are even (distinct but closely related) definitions of specific types of numbers in different foundations. But there is no general definition of number.
Or maybe I'm just pointing out that the idea of "numbers" speaks to a family resemblance. No single definition could hope to pin down "numbers" in some exact sense. But there is a constraints-based definition, or a structuralist definition, in that numbers are whatever it takes to get certain number-like operations - like those that preserve certain global symmetries, such as commutativity or associativity.
So you wouldn't expect a hard and fast definition of something general. As a foundational fact, all we can talk about is a family resemblance which emerges as we enforce greater and greater constraint in terms of bounding symmetries. Then in the limit, we arrive at associative division algebras.
So the octonions are not associative but they answer to the weaker constraint of being alternate. They still deal in numbers, but a slightly less constrained version.
The general definition is thus inherently fuzzy. The most highly constrained or specified kind of number system isn't broad enough to capture the weaker kinds that are possible with some of the constraints relaxed or even absent. And yet at some point the "general definition" is so weak that it doesn't then capture the essential operations that do define being part of that general family.
This is a familiar issue when framing scientific law. Philosophy of science says exactly the same thing.
So you may be agnostic about foundational approaches. But philosophy of maths can't afford to be. And I think the same understanding of the trade-off between the general and the particular applies across all the rational disciplines.
You are simply looking for the wrong kind of "general definition". Structualism is needed because that focuses on generic relations of which numbers then can be the various possible kinds of object.
Why wouldn't ZFC count?
Bill Clinton made that very same argument to try to wiggle out of a sex scandal. In the end he lost his license to practice law and was impeached (but not convicted).
Quoting tim wood
You and @apokrisis seem to feel that "a number is anything that's number-like" and "quantities is whatever can be quantified" represent valid definitions. What happens if the biologists get hold of this trick? A fish is whatever is fish-like. A cat is whatever is cat-like. A virus is whatever is virus-like. And the deepest question of all: life is whatever is life-like.
This sounds like a fast path to meaninglessness to me.
Quoting tim wood
No I'm not confused on that at all. There were cats long before a biologist said that a cat is "a small domesticated carnivorous mammal with soft fur, a short snout, and retractile claws." The thing clearly precedes its definition.
A definition is more like a classifier. I'm working in a factory and my job is to stand at a conveyor belt and throw things into one bin or the other: cat and not-cat. A definition is a set of criteria that let me unambiguously do that. The definition lets me recognize things that are cats; and things that are not-cats. In other words I'll throw the cats into the cat bin and the not-cats into the not-cat bin with as close to 100% accuracy as possible. That's what a definition is.
Now today my job is to identify quantities versus non-quantities. So my definition is, a quantity is anything that can be quantified. But that's no help! You've just given me a different syntactic form of the same word. You have NOT provided me with classification criteria. So "quantities are things that can be classified" is not a definition, nor is "a number is something that's number-like." You haven't told me how to sort the objects into the bins.
Quoting tim wood
A good definition does let me determine whether a given object is or isn't the thing in question. If an object comes down the conveyor belt and it's a small furry domestic animal etc., I know it's a cat.
Quoting tim wood
It's a strong description. It's a description that fits 100% of the things we wish to include, and none of the things we wish to exclude. If a description satisfies that criterion, it's a definition. "Quantities are things that can be quantified" is no help at all. It's a description but not a definition.
Quoting tim wood
To be sure, I have no idea what you mean by existence. I would say that i exists because it exists in math according to the formal rules; and also because we see many instantiations of the i in the physical world. That latter point isn't obvious to everyone but for example anytime you make a 90 degree counterclockwise turn, you are instantiating the number i in the world. And of course i comes up in physics and engineering all the time.
Quoting tim wood
Yes I agree with that. A definition is whatever you can write down on an index card for me that will allow me to recognize cats and numbers and quantities as they come down the conveyor belt. Functional. Good word for it.
Quoting tim wood
To sum up, or rather to get back to basics, you claimed that numbers represent quantities. The number i represents a phase angle in electromagnetism or a quarter turn if you're in the plane. But I don't see those as quantities. So I have to ask again, what is a quantity? Are you claiming that the number i represents a quantity? That I do not agree with. I don't see it.
There's no definition of number in ZFC. In ZFC we have a definition of natural numbers, and we can make definitions of the integers, rationals, reals, complex numbers quaternions, transfinite ordinals and cardinals, hyperreals, and many other types of number.
But there is no general definition of number. If a thing comes down the conveyor belt and I have to say if it's a number or not, of course I can identify the types of numbers I already know about: integers, reals, etc. But I can't determine in general what is a number. ZFC offers no help in this regard.
The quaternions are numbers whose multiplication is not commutative. The transfinite ordinals are numbers whose addition is not commutative. How weird is that, right?
Good idea and a very natural attempt; but arithmetic properties aren't sufficient. Weirder still, there are numbers that lose associativity as well, such as the octonions. Octonions come up in physics so these are not only of abstract mathematical interest.
I intend to go back to your first post on the subject and respond in detail to your comments on mathematical structuralism and category theory, so I hope you can be a little patient. I want to start at the chronological beginning of your posts on the subject and I can't do that tonight.
In short though, mathematical structuralism is more subtle than just listing arithmetic properties like associativity.The kinds of properties that they use in category theory are ... well, they're kind of weird and nonintuitive when you first see them. The structural relations they have in mind are various types of universal mapping properties. It's hard to do justice to what this means in a simplified format but I might take a run at it once I get into responding in detail to your earlier post on structuralism.
If you don't like the fact that numbers are defined in terms of set theory, and further properties deduced from there, I guess you won't like the fact that numbers are also defined in terms of field axioms.
And no, new types of number are not "discovered all the time".
I certainly can't understand how you would have gotten that impression. Many specific types of numbers are defined within set theory. But there is no general definition of what a number is in set theory or in any other foundational approach.
Quoting tom
Not a bad idea. But the field axioms don't say anything about numbers. It's true that many types of numbers satisfy the field axioms, such at the rationals, the reals, the complex numbers, the integers mod p, and all the finite fields of the form p^n.
However, the set of rational functions in one variable satisfies the field axioms, but rational functions are not numbers. Rational functions are quotients of polynomials. It's not hard to show that they can be added, subtracted, and multiplied. It's a standard, somewhat nontrivial exercise to show they can be divided. So the field axioms aren't sufficient to define what we mean by a number.
https://en.wikipedia.org/wiki/Rational_function
Quoting tom
The quaternions (discovered in 1843 by Hamilton), transfinite ordinals and cardinals (Cantor 1874-1890's), the p-adics (Hensel, 1897), and the hyperreals (Hewitt, 1948) are a few examples that come to mind. These are very recent developments in the history of math. People didn't used to believe in zero, negative numbers, rational numbers, real numbers, or complex numbers. Each time someone discovers a new type of number, mathematicians have to expand their own ideas about what constitutes a number.
Wasn't that my point. You can still be a numerical object even if you don't qualify for the full checklist of arithmetic properties. So the octonions lack a particular property. But they still count as part of the algebra family which takes numbers as their relational objects.
It probably helps that we can see why octonions lack this further constraint. We get the feeling they would express this property if only they could, as that is the general direction they are headed. But their own nature prevents fulfilling that goal. Therefore they qualify to be part of the family even if they can't tick every last box of some ideal definition.
Do you know much about Bourbaki structuralism - the three mother structures of algebra, topology and order - or category theory structuralism?
As I've been saying, you seem to want a definition founded on the mathematical objects rather than the mathematical relations or structures. But that just seems an antiquated notion.
And what is true in philosophy of maths is true of metaphysics generally. Nature is becoming understood in terms of its global structure rather than its local atoms.
That has been the recurring sticking point in any discussion we've had. You just presume the correctness of a reductionist or atomistic metaphysics. You then seem to have no understanding of the alternative view that is that of the structuralist, process philosopher or systems scientist.
Quoting fishfry
Maybe just focus on that. Structures are the objects now. Then morphism is how structures have a relational structure that allows acts of mapping.
So the essential property that founds the whole business is closure or symmetry. Hey, just like physics!
(And then I should add that the fundamental question becomes how could closure emerge? What constrains the openness that seems the alternative?)
I don't want a definition. I merely pointed out that there isn't one in math. You agree with this by now, yes?
I said you seem to think that a definition would be in terms of the mathematical objects involved, and not the structure of relations needed to produce them via a holistic system of constraints.
A reply that makes some contact with the relevant philosophy of maths would be appreciated if of course no longer expected.
I did not write the quote you attributed to me. What is your attitude problem?
I stated originally that there is no general definition of number in math. Nobody has provided a counterexample and you now seem to agree. That's all I said. I actually can't imagine why you are going on about structuralism, which has nothing to do with what I said.
I'm not making any point about philosophy. I'm making a statement about math. There is no general definition of number in math. This is uncontroversial and widely known. You are going off on wild tangents that don't bear on what I said and that don't falsify what I said. If you choose category theory as your foundation, there's still no general definition of number.
Perhaps you would consider starting a thread on mathematical structuralism. It's an interesting topic. It has nothing to do with what I said, which is that there is no general definition of number in math.
I get a bad attitude pretty fast when someone like you plays cute with a quote. If you leave off the important part of what my sentence said, that is flat out misrepresentation. Expect a swift kick in the arse.
Quoting fishfry
Keep trying the same trick. You are looking worse and worse.
Quoting fishfry
Your inability to discuss the foundations of maths is noted.
Your complete misunderstanding and lack of comprehension of category theory and mathematical structuralism was evident to several other posters the last time we discussed this. I was hoping in the limited time I have each day to post here that I would gradually work through your earlier posts in this thread and help you sort out some of your ideas. But you are simply too rude and annoying for me to bother any more.
I'm done responding to your posts on this site. It would be for the best if you'd simply stop responding to me. Regardless I will no longer respond to you.
Here we go. You and your circle of imaginary friends.
Quoting fishfry
Funny. I was waiting for you to start.
Forgive any apparent crude or unintelligible thoughts. As to the question of whether numbers exist:
Who can question whether number exists in the mind? Whether they have concreteness is simply to say, 'do they exist in reality as they exist in mind?' Obviously not. Temporal construction is such that inequality defines its nature. Equality, on the other hand, can only be outside of temporality. One might say: 'equality can only exist eternally.'
The question then arises, 'what is the nature of number?' Conjecturally, one might say, number is a series of equal values (quantity). Hence, Pythagoras' and other ancient mathematicians' inclination to render number as equal, whole values. If this is an accurate description of number, then it follows, the concept of number is tied to the idea of a 'unity' value (unit measure).
The question can then be asked, 'where does the concept of 'unity' come from? Again, conjecturally speaking, unity may only be understood as an eternal concept. So, the question of whether number exists, is tied to the answer to the question of whether eternity exists.
Yeah. The historical view is a good way to get at it. There was a reason why the Greeks were so horrified by the notion of an irrational number. That very reaction betrays the underlying belief about what a definition might be.
And so we have "oneness" as the central object of arithmetic. And we have "a dimensionless point" as the central object of geometry.
The Greeks were discovering what the maximally invariant mathematical objects looked like - the ones that had irreducible identity despite all possible operations that might attempt to change that in one of the mathematical families.
Having established the highest symmetry identity operations, then maths could progress by identifying and relaxing the various constraints that ensured the existence of these ideal objects - the 1 and the point.
Geometry could go non-euclidean and topological. Numbers could loosen to include negatives and irrationals.
Bourbaki's talk about the three mother structures makes me wonder where order structure fits in to the Ancient Greek story. I guess Aristotle's work on the logic of hierarchies - the [genus [species]] relation - does describe what is the most primitive notion of set theory.
Quoting cruffyd
I get what you mean but in that direction can lie hard Platonism. However the altermative I prefer is another long story.
Thanks for the mention.
My knowledge of mathematics and its history is only rudimentary. Perhaps I slightly misunderstand the role of the Pythagoreans in relation to irrational number. It seems, by most accounts, that the Pythagoreans were not 'horrified' by the idea of irrational number, but rather of this idea becoming generally known by the 'uninitiated.' By Hipparsus' own account, Pythagoreans may have been well aware of irrational numbers and accepted their existence.
To the extent that the idea behind whole numbers may be tied to an eternal, or extra-temporal idea, they fall into the same category as what we are calling 'irrational,' hence the question at the outset of this discussion. To the extent that a number such as pi can be concretely understood as to its function, it could be called 'rational', loosely speaking, of course.
Again, in returning to the question of 'existence', numbers have more existence as concept than as concrete. Defining existence itself can become problematic. One school of thought mentions as many as three other types of existence apart from the concrete, or physical, one of which is conceptual.
My remark was intended as lighthearted. What the meaning of "is" is was very big in American popular culture during that particular scandal. This is the only time I've heard that question raised since the Bubba and Monica affair. Cigars, stains on dresses.
If Americans want to know how we ended up with a monstrously crude man as President like Trump, I'd say the bar was set low when the American people made Bill Clinton a two term president and never held him accountable. Trump and Bubba used to be golfing buddies. You think they talked about women's rights?
I think my interest in American politics is off topic here so I'll let it go. But when you say it depends on what the meaning of "is" is, you can hardly be aurprised that the first thing anyone would think of is Bill Clinton and the intern. It's the 20 year anniversary of that scandal right now. So it's in the air.
Quoting tim wood
Ok. So when you say that a quantity is that which can be quantified, you are NOT saying that a cat is what which can be cat-like.
I confess to not understanding why anyone would regard this as a sensible response. But I'm sure that's more due to my philosophical ignorance. But if I'm ignorant, this would be a point where you could educate me. When you say, "A quantity is that which can be quantified," what actual information are you imparting? To me it just likes you changed the form of a word without adding meaning.
Quoting tim wood
We definitely disagree and I am curious to understand your reasoning. You said that a quantity is that which can be quantified. I don't recognize that as the answer to any question I asked. I'm sure the communication problems are all on my side, but I'd like to bridge them if that's possible.
Quoting tim wood
My university training is in mathematics, although my post-university career involved following math only at an amateur level online. But I absolutely regard i as a number. To me the number i is as concrete as the number 6. It just refers to something different than what 6 does.
6, you see, does generally represent a quantity. Six ducks in a row, six eggs in half a dozen, six bullet items in your PowerPoint slide. The number 6 is instantiated in everyone's every day experience all their life.
Now the number i, as it turn out, is every bit as pervasive and a normal part of our daily lives. However people don't recognize this, because the number i is taught very poorly in high schools around the world.
Forget that crap about "the square root of -1," which always sounds like bullshit because they just got through telling you that there is no square root of -1.
Think instead of i being a gadget that keeps track of how many counterclockwise turns of 90 degrees you make. Say you start facing east. You then turn north. Call that i. Then you turn west. You are now facing directly opposite the way you started. In other words ... i^2 = -1, and this notation is simply an expression of something very simple. If we make two quarter turns to the left, we are now facing in the exact opposite direction of where we started.
Now one more turn is -i, and one more turn aft that is ... 1.We're facing east and we just discuvered that i^4 = i^0 = 1.
So i is a number. but it is not a quantity. What it is, is an instruction to make a quarter turn left. That's what numbers can sometimes be. Representations of geometric transformations.
A general complex number is z = a + bi where a and b are real. An alternate and more insightful notation is polar representation. If z is a complex number then we can write z = re^(it) in complex exponential form, there t is the angle made by the line segement between the origin and z, and the positive x-axis.
In trig form this is the same as saying z = r(cos it, sin it). This rotates the oringinal vecor through and angle of t, and it scales it by a factor of r.
If you plug in t = pi/2 and r = 1you get the special case of z = i. In fact the case r = 1 is very important because as t goes from 0 to 2pi you get all the points on the unit circle.
So every time you turn left -- at a traffic intersection, on a street corner if you're walking. or if you're just standing in your living room spinning around conterclockswise: You are instantiating the complex number i. Every time you turn through an angle of t, you end up at a particular point on the unit circle.
That's not all. The number i is an essential part of modern physics and engineering. Having a symbolism for something being 90 degrees out of phase is very handy. So i can be defined in formal math, and it comes up in physics. It's a number, and it is instantiated in the world.
So YES, i is a number. But NO, i is not a quantity. The number 6 is a quantity. It's 6 of something. But i represents no quantity. I represents a quarter turn in the plane. And geometric rotations and scalings of the plane happen to have very algebraic properties.
You don't even need a magic "square root of -1" to do this. There's a particular subset of 2x2 matrices whose entries are real numbers. They are an isomorphic copy of the complex numbers. So nobody has to believe in anything "imaginary." If you believe in the real numbers, then you'll agree to believe in 4-tuples of numbers arranged in a 2x2 array, along with the usual array operations of matrix addition and multiplication.
One more example. The area of a circle with radius 1 is pi, right? Now is that a quantity? A quantity of what?
We determine the area in multivariable calculus by defining the two-dimensional Riemann sum. We fill up the circle with little squares and count the squares. Then we fill it in with smaller squares. At the end of that limiting process is the area of the circle, which comes out to pi.
But there's no quantity anymore. At each step there was a finite quantity of little squares. But in the limit, there are NOT infinitely many infinitesimal squares. Calculus abandoned that approach. Instead we just work with the limits. So at the end of this process, pi is a number but it's not a quantity of anything.
Quoting tim wood
I'm a math guy. Of course i is a number. I mentioned this earlier in a reply to @Tom, if you read back a few posts you might find it. I believe in the mathematical reality of all mathematical structures. [Note that this is not to say I believe in their physical reality. Only that if i can construct something in math, the it's a mathematical object and has mathematical existence. I make no general claims about the world].
Quoting tim wood
Well now THAT is the good question!! In math, nobody bothers to ask the question because it's a question of philosophy and not math.
In philosophy, we're seeing that it's damned hard to pin down what a number is. And it's fun to try. Or at least it SHOULD be fun to try. When it becomes less than fun I become less inclined to play.
Clearly SOME numbers represent quantities. Other numbers represent scaling and rotations in the plane. Ordinal numbers represent order types. Cardinals DO represent quantity!.See we even have two different notions of transfinite numbers, one that represents quantity (cardinals) and one that represents order (ordinals).
We have familiar numbers like pi where we're hard pressed to say what quantity of anything that represents. Pi is defined as a ratio, it's defined as an infinite series, it's defined as the smallest positive zero of the sine function, which we can define via an infinite series so that there's no geometry involved.
Some numbers represent quantities and others don't. So it's a tricky thing to accurately express what a number must be in general to be considered a number. Every rule anyone thinks of has lots of exceptions.
Quoting tim wood
I absolutely agree that i is a number. But it is not a quantity. You said it is a quantity. I want to know by what criteria to you call i a quantity. And it's wholly inadequate to say that i is a quantity because it can be quantified. Any fairminded philosopher must see this.
Perhaps this is pedantic, but even in terms of rotations in the complex plane i does have a couple of associated quantities with its notion of multiplication. It represents an anti-clockwise rotation of 90 degrees and a magnitude of 1 in terms of the size of complex numbers. Even though it doesn't represent the same kind of quantity as scalars, there there are two associated magnitudes when thought of in terms of rotation and scaling of a point's position in the complex plane. You probably already know this.
Another good example, which I believe you brought up, is that the integers mod p where p is prime form a field under multiplication mod p and addition mod p- the algebraic structure (with 0 removed wherever it's mathematically appropriate) which is most similar to folk intuitions of how numbers should work. But the idea of division as multiplication by multiplicative inverse mod p is nothing like the idea of division present in intuitions for fractions (rationals) and reals (rationals + irrationals).
The reals (excluding weird stuff about 0) under multiplication and addition in the usual sense satisfy modern intuitions about what it means to be a number. When those intuitions are formalised, in turns out that there are other structures which aren't commensurate with folk intuitions that nevertheless satisfy the axiomatisation of a field inspired by those folk intuitions.
Another wrinkle is introduced by the idea of an isomorphism. Say we have the set of numbers {0,1,2,3,4,5,6}, and addition and multiplication are equal to their remainders upon division by 7 (this is the integers mod p). We have seven elements, and since they satisfy the rules for addition and multiplication and consist of numeric symbols, it would be a stretch not to call the elements of this structure numbers.
However, relabelling 0=A,1=B,2=C,3=D,4=E,5=F,6=G, the laws of arithmetic and the sense of equality being equality in remainder when divided by 7 produce curious statements like:
A*x = A where x is in {A,...,G}
A+x=x where x is in {A,...,G}
B*F=C*G, which is equivalent to 1*5=2*6=12=5
C^-1 = E (which states that 2*4=4*2=1)
If presented with the {A,...,G} representation of these numbers, someone who hasn't been trained in mathematics probably will not recognise the manipulations as equivalent to manipulations of integers modulo p. Are they still numbers? Mathematically they're equivalent to numbers.
You could do the opposite trick by labelling the symmetries of a square as numbers. Four reflection symmetries, three rotational symmetries... Further tricks by identifying usual group structures (like the integers under addition modulo 7 while ignoring multiplication) as their corresponding symmetric group representations...
Really all this says is that 'what is a number' and 'do numbers exist' are to some extent independent from the concerns of doing mathematics. What matters is that the math functions as it's set up to.
Another way of putting it: would a change in the ontological status of numbers change either the truth or the sense of 1+1=2? What about the ontological status of rotation groups: would what you believe about the ontological status of rotations change anything about the idea that if you rotate an object 90 degrees 4 times, you may as well have not rotated it at all? I don't think so, in either case.
No. It's not. That's the point. i is a number but it's not a quantity. That's a counterexample to your idea that a number is something that is a quantity or that can be quantified. Simple as that.
I pointed out that it's very difficult to define in general what a number is. You suggested that a number is something that can be quantified or that represents or is a quantity. I gave as a counterexample the number i, which is a number but is not and does not represent a quantity.
You said a quantity is something that can be quantified. I don't find that helpful because it doesn't tell me what a quantity is. If you tell me a cat is a furry domesticated mammal with retractile claws, that's a lot more helpful than saying that a cat is anything that's cat-like.
That's right. I noted that there is no general definition of number in mathematics. A well-known and true observation. For whatever reason, this simple and harmless statement triggered several people. I still don't understand why.
I do of course agree with you point that 2i is a quantity of two i's, like 2 apples is a quantity. So the question reduces to asking exactly what is a quantity. @tim wood brought up the idea of quantity a while back so I asked him what is a quantity, and so far I have not gotten an answer.
But ordinals I think are the best example of numbers that absolutely can not ever be interpreted as quantities, since the same cardinal can be rearranged to represent many different ordinals.
Number is not the same as quantity. I think that's clear.
Quoting fdrake
Right. And some structures that satisfy the field axioms are most definitely NOT numbers, such as the rational functions with coefficients in a field.
I'm not entirely sure I understood the theme or message of your post. All I'm saying is that there's no general definition of number in math; and even for logicians and philosophers, it's very difficult to pin down what a number is. I've never seen a successful definition.
Either people were triggered or they thought there are some good approaches worth discussing in philosophy of maths.
Quoting fdrake
Fdrake is right. If we want to ask what i quantifies, it quantifies the number of dimensions that a number is constrained by. So i is a widget to rotate a real number into an orthogonal direction that turns the number line into a number plane.
The number line stands for the most constrained notion of continuity. Complex numbers relaxes that strong constraint and allow numbers to wander in two dimensions. And the numbers still behave like numbers - objects that meet the functional criteria of associative division algebras.
We can continue to relax the number of dimensions in play. We could consider a three dimensional number. But now it doesn’t behave arithmetically. It is not a suitable object of algebraic structure - a fact of undoubted physical significance when it comes to why space winds up being three dimensional.
Then with quarternions, we have four dimensions and a bounce back to a large amount of algebraic structure. Five, six and seven dimension again see that structure disappear. Then the octonions provide a last echo.
So i is a good example of structualism at work. We can define some basic relational properties that numbers are meant to have. The associative division algebras do that. And then we can see how the “hard structure” emerges as constraints are added.
As we constrain the dimensionality that defines the continuous space in which discrete mathematical objects are meant to move, we can see the role those constraints play in actually defining the mathematical properties those objects are understood to have.
The limits maketh the objects and not the other way round.
By saying that the magnitude of i is 1, what I meant wasn't that there was a single i, an answer to how many 'i's are there - but that a vector that starts at the origin in the complex plane and points to i has length 1. This allows for there to be real numbers of 'i' as the 'number of i's in a complex number, so to speak.
More generally, speaking about complex numbers like z=4+pi*i. Pi isn't exactly an answer to 'how many' i's are there in z since its interpretation is severed from counting numbers in a few ways. The first way it's severed is that z is not a multiple of i in anything like the counting number sense (there are pi-4i i's in z), so we cannot chunk z into i sized bits through division. an 'i sized bit' doesn't even make sense as imaginary numbers don't enter into the notion of size for complex or imaginary numbers.**
The second way the interpretation of the magnitude of z is severed from the interpretation of a real number or fraction is that z has two senses of magnitude inherent in it. There's the real part and the imaginary part (which individually work exactly the same as real numbers and usual counting in terms of 'how many' questions, to the extent that irrational numbers can be said to be answers to 'how many' questions) or there's the polar form of the radius and angle - requiring two descriptors of magnitude to specify the quantity ('number') rather than the single one for scalars. Polar form and Cartesian form for complex numbers also have differences in interpretation since the polar form contains an unbounded quantity (radius) and a bounded one (the angle), and Cartesian form is done in terms of two unbounded quantities (the magnitudes of real and imaginary parts). They also mean different things (polar form and Cartesian form) even though they are just different ways of talking about the same thing (naming complex numbers).
There's also the wrinkle which you already mentioned about the tension between irrational numbers (which are implicated in the magnitude of complex numbers in both directional and radial senses) as magnitudes and fractions as answers to 'how many x go into y' questions. Even the Gaussian integers have this problem (such as 1+i having magnitude 2^(1/2)).
Actually looking at 'numbers', even in relatively simple cases like these, shows that there's no single sense of magnitude or quantity implicated within them - even if there are formally equivalent representations.
**the closest approximation to this in the complex plane being dividing a complex number z=x+iy by r=sqrt(x^2+y^2) yielding u=z/r, u has magnitude 1, which is the same magnitude as i - all this says is that z lies on the unit circle and u can be obtained again by scaling by r.
This topic is simply too vast for me and I personally try not to think about it too much, lol.
There are too many things going on here but I would like to start with an inquiry on the above statement: specifically is it Peircean or not? I haven't found anywhere Peirce expressed atheism or the like. Or maybe I didn't follow you correctly.
The next maybe another inquiry about your support of "symmetry" concept, but let's set it aside for a moment.
It is notoriously difficult to agree what Peirce actually believed about god or divinity. But he himself stressed he certainly did not follow any kind of orthodox view.
And my point there was that he definitely did not argue for an external creator with some mission in mind for mankind. Instead, he identified the divine with the vague ground of being - the Firstness of pure unformed potential. And so the Comos is a state of logical regularity that evolved into being in a purely self-creating fashion with no purpose in mind except to be "increasingly reasonable" in its lawfulness and organisation.
He did say he was more Buddhist on this score. :)
wow, if you happen to have a public link, kindly share. I found only this content http://www.gnusystems.ca/CSPgod.htm#aq1 ("I think we must regard Creative Activity as an inseparable attribute of God." C.S Peirce.)... there maybe more pieces but let another time to connect them together.
Just to play fair with the thread, numbers are in mathematics which is in turn sub-semiotics. I am not sure the last has been maturely explored but math is thought by Platonists as another world. Even the "semeiotic" sounds much to do back with Plato's Ideas, now with a better weapon of Synechism.
If the multiverse-like metaphysics is accepted then we can perceive such a Cosmos that circumscribes it in. Old story while I find your posts more interesting and will switch to enquire about Peircean buddism or non-Peircean symmetry, where possible.
For symmetry, I am not sure you have come across talks similar to these https://www.closertotruth.com/series/why-do-we-search-symmetry
I drew a note that fundamentally deep down, symmetry is quite empirical and approximate. It's useful in many talks but we have to recognize its limitations and avoid it at extremes.
Being aware of no symmetry from Peirce, I think if we still need to linger on it, we may want to analyze Synechism, not only Tychism.
Actually the quote I was thinking of was misleading as it wasn't connected to his evolutionary cosmology but to the more mundane thing of how his Christian contemporaries view his "scandalous affair". Buddhism wouldn't be so judgemental.
So it was more that Eastern metaphysics was in the air in his time as something exotic, but not really studied.
Here is a more direct reference in terms of his evolutionary cosmology where he talks about its roots...
Quoting Dzung
Yeah, I don't think Peirce said much about symmetry and symmetry-breaking principles. It was implicit rather than explicit at best.
Peirce had a Victorian level understanding of phase transitions and other physical manifestations of symmetry breaking. Group theory and its fundamentality in physics was a 20th century thing, after all.
How can you identify types of numbers if you don't know what a number is?
ARTHUR W. BURKS - PEIRCE'S EVOLUTIONARY PRAGMATIC IDEALISM
https://deepblue.lib.umich.edu/bitstream/handle/2027.42/43816/11229_2004_Article_BF00413590.pdf?sequence=1
in deed your above quote is from a letter of him to Williams James. The majority of his quotes are scattered over different kinds of media but I do think all are connected, much like his philosophy about the continuum - Synechism. He also said "I do not agree with you that my papers about the evolution of the Laws of Nature are the best things I have done."[/i] (C.S.Peirce) and "I think unquestionably my best work has been my Logic.". This really helped me to grab a knot from his web.
The information conveyed in any of his works is massive and cannot be plainly elaborated in a small article or even book. It seems his doctrines such as triadic reduction, synechism, infinitestimal and even his flatly established religion (in the same letter he also mentioned people had scoffed at his religion so he would refrain from expressing it)...have yet to be duly understood.
If I have an orange, and beside it I have another orange. What I have is an orange and an orange. The concept of '2' oranges is one I apply. The '2' does not exist outside my mind in the real world. If you say it does, where is it? Show me it. There is no '2' in the objective world, only matter and energy. It's easier to see that '0' does not exist, but '2' is no different. Thus Maths is invented. It is a spectacularly accurate tool for describing the laws of nature, but it is created by us just as any other language is.
This poses a number of questions-
1. Is the algorithm that generates the digits real? If so, are not the digits also real?
2. Does number precede geometry or vise versa? Some extraordinary infinite series have been discovered that map, with infinite precision, real numbers like Pi, e, etc.
If infinite series can map Pi exactly does that mean that number precedes geometry (space)? Do numbers exist in God's Mind before space or do numbers arise out of (Euclidean) space?
It seems to me that number is more primitive than space and as such they precede space.
All numbers are directed movements.
To me the way to frame this question from the very beginning is how numbers exist, because if they don't exist in some way then what are we talking about in the first place?
Numbers are directed movements in the finite sense, where the absolute nature of number occurring through "infinite directed movement" as a constant limit.
Infinite movement, as perceivable no movement, can be equated to a wheel spinning at the rate of infinity where while it is moving is observed as "still" and "constant"; hence number has a dual state of absolute truth and relative finite truth (with finiteness being multiple infinities).
I apologize for the long post ahead of time, however in this case it may be a necessary "evil".
We see "number" in the universe relative to the symbolic context in which we apply it, for the symbol acts as a medial point between the observer and what is being measured.
The problem occurs in the respect of the symbol itself and not just interpretation, but how it reflects "our" perception of reality and in these respects takes on a subjective context, that while objective in many circumstances, does not necessarily mirror the objective nature of reality we observe it through.
The question of "perceiving" a number, as an empirical entity (with empiricism being founded in directed movement), is a question of observing not just directed movement but universal symbols that reflect that directed movement.
Considering the linear nature of time necessitates a 1 dimensional nature, where this 1 dimensional nature effectively observes "1 as directed movement" through the line, it may be logically argued that the 1 dimensional line and "1" are both the same literally and symbolically.
We can observe this in the quantification of any temporal object is fundamentally an observation of time and numerical with number having a relativistic nature of part through part.
So if I see 1 orange, I see one direction in time.
If I see two oranges I see 2 times zones, or two directions in time, where these 2 directions in time still exist as 1 direction considering this is "one" 2 (if you understand what I am saying here).
So while time may be linear, but the line exists relative to other lines with these multiple lines observing multiple directions which may be off by just a quantum of a degree, time as linear results in time as multdirection effectively leading to a circle or sphere as "all directions" as 1. In these respects 1 takes on a dual role of constant and absolute truth through the "Monad" while observing a relativistic nature of "Monads"(atoms) that again exist through linear directions in themselves.
The nature of number alternates between a relativistic notion and one of absolute truth, where each finite reality is but an extension (or approximation) of an infinite one.
So if we quantify all of reality as "one" we are left with instead of a line, a 1d point existing as pure movement. The point exists through a point as a point and in turn can only be observed approximately as a boundless field in one respect while the connection of the points existing through eachother through lines without direction (negative dimensional).
Then you have the question of frequencies as literal numbers being alternating lines as a 1 dimensional line inverts to another.
So an angle observes 2 directions as 1 direction in the respect the angle is still directed and exists as a line in itself when viewed from far enough away. The concept of the "degree" which all angles are composed of becomes relativistic as a degree is strictly the number of geometric shapes which fit in a circle.
The foundation of the "degree" as a relation of geometric forms.
1) The circle is the universal form through which all forms exist.
x) The triangle, as three points, exists 120 times within a circle of 360 degrees with each point acting as a degree in itself. Hence as 120 times the angles which form the interior of the triangle (from the center point) form the interior of the triangle as 120 degrees.
2) The square, as four points, exists 90 times within a circle of 360 degrees with each point acting as a degree in itself. Hence as 90 times the angles which form the interior of the square exist as internal 90 degrees.
3) The pentagon, as five points, exists 72 times within a circle of 360 degrees with each point acting as a degree in itself. Hence as 72 times the angles which form the interior of the pentagon exist as internal 72 degrees.
4) The hexagon exists 60 times with an internal degree of 60.
5) The septagon exists 51.4287 times with an internal degree of the same.
6) The octagon exists 45 times with an internal degree of the same.
7) The nonagon exists 40 times with an internal degree of the same.
8) The Decagon exists 36 times with an internal degree of the same.
9) The 1 directional line exists 360 times as 1 degree with the 2 directional line existing 180 times as an observation of 180 degrees.
All degree, through angulature, exists as relation and is subject to the number of relations measured, hence the degree changes with the number of "x" shapes applied to the circle. Measurement itself is relativistic.
Yet the degree is still a line and is 1 dimensional, so what we understand of the number as a line is strictly 1 as relative units.
The frequency, in the respect it is composed of multiple alternating lines within a give framework is still projection in one direction as well, with the frequency appearing as a 1 dimensional line from a different framework. The 1 dimensional line can be observed as a quantum frequency necessitating all "1's" are composed of a finite set of numbers in themselves where "relatively speaking" a "1" may not be the same to another "1" as the first 1 may be composed of 1/1, 2/2, 3/3 to infinity and the second one may be equal to (1±x)/(1±x), (2±x)/(2±x), (3±x)/(3±x) to infinity.
Curvature equates strictly a series of approximate angles, which appear as angles relative to some limit of a different size.
So while reality observes number in a literal sense, because an localization results in a simultaneous clarity and ambiguity number takes on a possibilistic, potential and random (approximate) sense as well.
Number exists as
1) a causal (with cause being structure) and random duality.
2) actualized locality (part or atom) and potential locality.
3) limit (directed movement) and possible limit (no-limit as no directed movement).
This argument may seem a little ambiguous because of the large amount of information in one section, and may be elaborated on.
In simpler terms, "number" is perpetually moving and hence because it is perpetually moving it is constant, but relatively ambiguous at the same time when we localize any phenomenon. While we may be able to continual quantify number not all number is quantifiable relative to time.
For example the abstraction of 1/2 applied to an empirical object results in the object being divided in half. The reverse is true as well, the object divided at center point, results in 1 object moving to two objects. The 1 objective represents a prior unity in time, a potential one in the respect it contains some degree of formlessness, with the 2 objects stemming from it as actual (directed and moving in time).
Now stepping back and looking at the time line in itself as a framework, and observing the 1 object moving to two objects, it can be observed three objects exist as 1 time line (1 object, 2 objects each 1/2 of the prior object).
The nature of time takes on a quantitative role in these regards where the 1 object is a timeline, the 2 objects as 1/2 each of the original, as time lines as well.
The division in the timeline observes a point of one object inverting to many.
Time is its own measuring system, and strictly is directed movement...nothing more or less. The application of time creates new time zones, where each object acts as a time zone in itself as itself.
However, the difference between abstract and empirical phenomenon leads to some problems when both are observed as both directed and moving.
All arguments are variations of the same thing, but differ due to the entropy of language.
Numbers as pure abstractions cannot exist without an empirical base from which the abstraction arises from, hence the number takes on a directional quality due to the temporal nature of all empirical phenomenon.
In turn all empirical phenomenon formed by number, take for example one using abstract mathematical concepts to form a building, shows that abstraction exists through physicality.
Now 1 as a continuous function can be observed in the respect 1 is defined through the function of addition/subtraction/multiplication/division as well considering:
1 = 3-2,4-3,5-4 to infinity or -2+3, -3+4, -4+5 to infinity
1= (1/2)/(1/2), (1/3)/(1/3), (1/4)/(1/4) to infinity or 2*1/2, 3*1/3, 4*1/4 to infinity
and so on and so forth. 1 is equivalent to continuous addition, subtraction, multiplication and division where 1 is equivalent to an operation in itself as the operation is a constant. 1 takes on a role of function as well as form in these respects as well as being composed of an infinite series of numbers through which it exists.
Here is a response I put on the material as a medial thread but it applies as well here, where 1 is equivalent to a continuous function.
1 is a function through the line, hence 1 is a equivalent to a process of directed movement where the line and 1 are the same through Pi.
All fractals are composed of further fractals as evidence by Pi.
1) Pi is: the symbol ? denoting the ratio of the circumference of a circle to its diameter
b : the ratio itself : a transcendental number having a value rounded to eight decimal places of 3.14159265
http://www.bing.com/search?q=Pi+definit ... 1B982EA403
2) Pi is a line between two points that exists from the center point of the circle to the circumference. All lines in turn exists as center points of a circle towards is circumference where all lines exist as the ratio of Pi as 3.14159...
3) The line as composed of infinite points is composed of infinite lines, hence the line is composed of infinite circles as all lines exist as Pi.
4) The line is composed as infinite circles projecting, hence the line is equivalent not just to infinite points but infinite quantum circles as well.
5) Each line, as composed of infinite further lines, is composed of infinite "pi's" where the line as Pi is composed of further Pi's. Hence Pi is divided by an infinite number of Pi being divided by Pi. All functions exists through further functions as 1 function, hence 1 is equivalent to a function that is a continuum. 1 is a continuous function.
Hence Pi dividing itself observes Pi as its own function of self-division conducive to 1 through the line where 1 is Pi as a function of perpetual self division.
f(x)= 3.14159?(x??)
............f(x)= (3.14159?(x??) =1
................f(x)= (3.14159?(x??)
.........................f(x)=...
or
f(x)= (3.14159?(x??))/( f(x)= (3.14159?(x??))/(f(x)= (3.14159?(x??))/…)) = 1
X= a continuous series to infinity where the counting of Pi has stop. X= the limit of Pi as a finite rounded number.
Hence “x = all number with all number equivalent to 1.”
"How often do you masturbate each day?
a. Twice
b. Three times
c. More often?"
1. All axioms are taken as self-evident, and as self-evident have a completely subjective nature where one person can see an axiom and see one thing, while another person can see another axiom and see another.
2. All axioms have an element of randomness to them where all axioms effectively mean nothing in themselves and are determined by the frameworks (equations, proofs, algorithms, number line, etc.) that determine them due to this subjective nature.
3. These frameworks, as axioms, are still subjective and in these respects contain an element of randomness in themselves based on point 1.