What is logic? How is it that it is so useful?
Logic seems to be a sort of ordering/rules. It is to make a set of objects or tokens (fiat symbols lets say) become consistently ordered such that an output will occur.
Its usefulness is a bit mysterious, but ordering physical objects/materials is the basis for modern electronics and computers. Turning the physical medium of electrical signals through copper/fiber optics or other substrate can lead to billions of on/off switching that is the basis of Boolean logic.
The math behind any engineering is obviously based on some sort of logic. What then is this logic? How does it metaphysically adhere to reality? How does it epistemically manifest in humans? And how is it that it can be wielded with such power to create technology in the sciences, engineering, and technology?
I guess the main question is what is the nature of logic and how come it is that its nature is so useful to humans?
@fdrake I thought you might like this question.
Its usefulness is a bit mysterious, but ordering physical objects/materials is the basis for modern electronics and computers. Turning the physical medium of electrical signals through copper/fiber optics or other substrate can lead to billions of on/off switching that is the basis of Boolean logic.
The math behind any engineering is obviously based on some sort of logic. What then is this logic? How does it metaphysically adhere to reality? How does it epistemically manifest in humans? And how is it that it can be wielded with such power to create technology in the sciences, engineering, and technology?
I guess the main question is what is the nature of logic and how come it is that its nature is so useful to humans?
@fdrake I thought you might like this question.
Comments (109)
Over on one side we have the world, and over on the other side we have logic. Isn't it astonishing that logic somehow just happens to "fit" the world?
But logic and the world are not so distinct.
It's like being astonished that a glove just happens to have five fingers.
How do I get outside of logic to ask about it?
Aren't I using logic to talk about it?
My brain hurts.
Did I just not ask that question?
I thought the OP introduced the idea.
I am not interested in making fun of it. It is the outline of many interesting things.
My question was not rhetorical.
Where does the OP introduce "getting outside of logic"?
"Wow, isn't it astonishing that logic just happens to fit the world..."
No; logic was chosen and grown within the world in order to set out how the world is. That's not astonishing.
Quoting schopenhauer1
is to look at it as something that is not necessary in every instance. And that point of departure is "outside" of accepting the process as necessary as such.
From what?
But what are you implying here?
What is the implication then?
That's because trying to explain logic, is like asking why 'why' means why. Logic is the basis of explanation, it is not 'what is explained'. If you want to understand logic, you understand how to use it, not why it is applicable or what it is. SO - I endorse Valentinus' response.
Without identity, what would these comments mean?
Indeed, meaning presupposes identity.
That's not to say these comments are all meaningful, but, hey... :)
:ok:
All that needs to be kept in mind is that logic [s]can[/s] must change according to what new discoveries we make. In fact, according to quantum physics the law of non-contradiction doesn't apply or something like that.
We’re not exactly the most logical creatures and really have problems dealing with abstract ideas.
The three laws are:
- Law of Identity
- Law of Contradiction
- Law of Excluded Middle
I was referring to the Law of Contradiction in another thread addressing the absurdity of starting from a contradictory position as if it is of use toward claims of truth and/or validity. A trick many mystics use is to lay out a contradiction as a premise or proposition alongside another equally absurd beginning in order to make them appear of value - the dogma being to accept the statement blindly and follow logically from a non-position and thus allowing them to arrive at any conclusion they wish to. It is the endless problem of inference that is also used to bolster such delusions.
The mystic roosts his argument in absolute truth, the scientist roots his argument in empirical and universal truth, while the post-modern philosopher, that's you, roots his argument in contradictions and opinions such as "there are no absolute truths," which is either absolute true or not true, or it is true sometimes, in which case there are absolute truths. if it is absolutely true that there are no absolute truths, there is an absolute truth and a contradiction ensues. And if it isn't absolutely true that there are no absolute truths, then it is universally or relatively true that there are no absolute truths and the potential for absolute truths to exist is still possible. So, what in the f*ck are you talking about?
As an example, there are infinitely many possible syllogisms, 256 distinct forms, 24 of which are considered valid in traditional logic, and 15 in modern logic.
What a logician calls logic are just those forms or patterns that are deemed useful for certain purposes in the space of all possible forms and patterns. Understood this way, it's not mysterious that we distinguish patterns that help us achieve our purposes from those that don't.
This is sort of going in the direction I trying to go. idea here is also leaning in that direction to. are looking in that direction.
I notice most of the answers here have to do with logic's place as already useful. I find it interesting that an inquiry on the nature or origin of logic is almost considered impossible. What is the implication of that then? Well, you can just say, "It's foundatioanal", and "it is what it is", but how unphilosophical is that? Here we have a set of tools that we use in the world to create other tools, but we don't and refuse to look at it closely?
Is logic something that the universe provides? Are we divining/discovering logic? If so, is logic just how the universe operates? If so, is this different than the idea that we are divining/discovering math? Is that the same thing being that math is also an ordering/pattern principle? Is it more foundational or less foundational then math then as it might underride math (pace early Bertrand Russell).
If math is simply something that is nominal- we make it up to help make sense of the world, why can it be used so effectively in things like generating outputs from inputs? If put to use in a technological context, it is the basis for modern engineering, science, and technology.
I think it’s a mistake to think that maths or logic is something that can be explained. The same goes for scientific laws. All of these are the constituents of rational explanation, not things that themselves can be explained. And it’s not unphilosophical in the least to recognise that, it's simply an acknowledgement of the nature of knowledge.
You can’t ask why the law of identity holds, or why elementary arithmetic proofs are valid. They are the basis on which judgements of validity are made. The issue that I think you're grappling with, is that there's a tendency in analytical philosophy to believe that there's a naturalist explanation for such elements - but all such accounts are circular in my view. Why? Because:
Quoting Valentinus
Say you try an explain logic in terms of neurology, of what the brain does (which is the stance of brain-mind identity theorists). You're obviously not going to see the elements of logic in literal neurological data. What you might be able to do, is infer something about the way the brain processes sensory data and follows rules - but as we all know, it's an extremely complicated matter involving many levels of abstraction. But in that kind of analysis, you're already utilizing the very thing you're wishing to explain. Logic (etc) are internal to the act of thinking. So you can't stand outside of it, and see how it works; you only understand how it works by using it, by applying logic. But you don't get to turn around and see its source. (That is the reflexivity problem of the 'eye not seeing itself'.)
I read a great quote earlier this week - 'The ‘world’ of experience is not given in experience: it is constructed by thought from the data of sense.’ C. I. Lewis (Mind and the World Order: Outline of a Theory of Knowledge, p29). And that is why maths is predictive - because it helps us to order and predict experience. But maths is not given in experience - again, it is constitutive of the rational operations of the mind. But because of naturalism, we're (usually unconsciously) seeing it in terms of the world 'out there' and the mind 'in here'. And that is the basic root of the problem - we instinctively believe it is (like everything) a result or consequence of evolutionary processes, that it's something that evolves. But the theory of evolution, or any other theory, already depends on logic, as it is built throughout on rational inference. We use it, and take it for granted as something we can explain, but actually, we can't explain it; it is the basis of explanation, not the subject of it.
So it needs to be understood that logic is not so much explained by naturalism, as what naturalism draws on, in order to explain; but that in so doing, it's actually not explaining itself. Once that is understood, a myriad of problems of modern philosophy are dissolved.
We get our percepts from the world, and then we order them, build concepts, theories, models, which we then apply back to the world from which they derived (making predictions mostly I guess), sometimes successfully and sometimes not. I don't think it is logic that fits the world, because logic says nothing about the world. Logic is our ability to relate and order what the world gives us.
Logic is essentially about relationships - differences considered together. Validity, for example, cannot apply to one premise, you have to have at least two and to relate them simultaneously to perceive if the conclusion follows.
It's as if one were to tune into the radio and remark that, based on your extensive sampling of radio, there seem to be alot of musical acts worthy of getting played on the radio. But of course you don't hear about the failures - precisely because they are failures. And math and logic is full of utter rubbish which we disregard for the same reasons. @Andrew M's point speaks to this quite nicely.
Husserl focuses on refutations of psychologism. Maybe that would be an area where you could focus some attention? I’m only one third through the book and plan to start reading Naming and Necessity soon too in order to, hopefully, compliment it.
I’ve never called myself a philosopher. In one sense I guess I’m an ‘amateur philosopher’ in the sense of “philosopher” as one who has studied the texts of philosophers. Post modernist? Not at all.
You still don’t seem to understand that distinction between “fact” and “truth”. Science doesn’t deal in “truths” unless you’re framing ‘logic’ as a ‘science’ - which is fair enough. Mysticism isn’t based in logic so it doesn’t have a higher claim to ‘truth’ over logic. Nothing, by definition, does.
Take it up in the appropriate thread.
What kind of question is it? You've clearly perceived this as an example of something general, and I'm interested in that general category.
This is Boolean logic, and it has little application to humans or to human life except when we want to consider digital electronics. If you mean to refer to logic in a wider sense, then this answer is unhelpful, and I apologise. :smile:
Boolean logic allows us to design networks of logic gates, and thereby to design computers, and the like. Pretty boring stuff, unless (like me) you spent your professional life designing digital hardware and firmware. :wink: Then it can be fascinating!
Quoting Wayfarer
The world is intelligible only because of identity and difference. The inherent logic of identity and difference grows out of our ability to distinguish one thing from another, which is obviously essential to survival and is necessary for any intelligibility at all. In our parsing of the world in terms of distinct entities the law of contradiction is inherent: a thing cannot both be and not be itself. This logic is also inherent in the law of the excluded middle; a thing is either itself or it is not. Mathematics as elaborated is not "given in experience", but on account of identity and difference, number, which is the basis of all our elaborated mathematics, is.
So, the world of our experience is logically in accordance with those three laws and mathematics, the only exception being what is observed in quantum physics experiments. But we have evolved in the "macro world" of phenomenal experience, so we should not be surprised if the "micro world of QM yields counter-intuitive results.
We notice that some arguments are truth-preserving, and we call those arguments logical. What further explanation of their nature or origin would be needed?
As an analogy, consider that the major encryption schemes that underlie internet transactions rely on prime factorization. But what is the nature and origin of prime numbers such that they are so special? Simply that primes are those natural numbers that have a specific characteristic that can be exploited for encryption (namely, that they are not divisible by smaller natural numbers). Similarly, logical arguments - those with the characteristic of preserving the truth of their premises - can be exploited for various things, such as building computers, solving problems and increasing knowledge.
Of course, those definitions don't exhaust what can be learned about prime numbers or logical arguments. But I think it shows that they can be understood as perfectly natural features of the world and not as intrinsically mysterious or other-worldly.
Quoting schopenhauer1
The traditional view would be that thought, language and the world are isomorphic, that the world itself has a logical structure that can be discerned. The modern view would be that logic is about the form of sentences, not their content. Physics, understood as applied math, would seem to locate form in the world again as suggested by slogans such as information is physical.
I believe that logic, an internal-mind system is secondary to the external world and derived from it. I don't know if you believe in evolution but if you do then it must be that our minds, everything in it, including logic, must be mappable to external reality. We wouldn't survive and we wouldn't be able to pass on our genes if we were illogical. Of course there are some ''illogical'' thought patterns that actually help us survive but that's another topic.
So you see, the question ''why logic works/exists?'' is explained and there isn't anything mysterious going on.
That out of the way it's important to note that logic works and is probably confined to a human scale - the part of the universe we're capable of perceiving. Some say that contradictions, impossible at the human scale, do occur at the quantum level.
Thermodynamics, Maxwell's equations, all sorts of electrical, chemical, and physical laws, these are the basis for much of the technology we use. These patterns of nature (i.e. laws of nature) are in-built into the system. They are not things we invented (pace a realism of some kind).
Of course, the way these patterns lead to more complex patterns (pace an emergentism of some kind), it not fully known. How combinations of patterns create systems that are more than its parts, are perhaps the most vague part of the process. We know the less complex patterns. We know perhaps, how these patterns can combine, and we know the output of more complex patterns, but how this emerging process really works in terms of less complex to more complex is harder to nail down.
But lets say one of the complexities out the less complex patterns is the pattern of evolutionary change in animal species. There is a pattern, perhaps, to how species respond to environmental stimuli and external pressures. These patterns in evolution produce patterns of behavior. Patterns for various species are conserved in what we colloquially call instinctual behaviors- ones that can produce outputs favorable for survival in a certain morphological/biological/ecological niche for that species.
Human evolutionary pressures resulted in a more plasticity. The plasticity allows for accumulated cultural learning. In this learning process, abilities to see the very patterns that compose the human, nature itself, and the very reason they can learn, are employed to recognize patterns, use linguistic encoding to symbolically represent those patterns, and then use those patterns to help in survival, find more comfort, and entertain ourselves. The superstructures of culture, institutions, and the like help glue together the accumulated cultural knowledge and learning and reinforce it in a kind of feedback loop.
Thus logic that is metaphysically composed of "natural laws" becomes logic that is creatures composed of the patterns, recognizing the very patterns they are composed of. This might be where @Banno was coming from in his idea that logic fits too well- like questioning why a glove fits so well. This also leans towards the idea that logic and math is in fact discovered.
Now, a counter of this is @StreetlightX objection that humans don't just discover natural laws, and laws that lead to technology, but other laws that are not useful in any way outside their own contained system. These are non-thermodynamic, non-Maxwell's equations, non-quantum theory, non-Boolean algebra, etc. These are mathematical systems that are fiat, made up, but are wholly functional systems in and of themselves, without mapping to any real world phenomena. In this regard, it is the pattern-finding that is primary. The output need not be useful or map to anything real. If I was to use an analogy with other animal traits, I might use the example of certain birds that reach out and move their beaks and neck to catch an egg before it rolls away. It might do this in circumstances that mimic an egg rolling. It didn't do anything "real" towards the eggs, but it kept doing its instinctual response anyways. Well, if humans are naturally problem-solving pattern-finders, this can be simply taken out of its original context for survival purposes as an exaptation of sorts. It is an ability that is there as byproduct of having the enormous amount of plasticity needed for original environmental pattern-seeking and problem-solving.
Both of you seem to be hitting at my last post.
Agreed. It's very much like how life itself evolves. A trait, in this case logical ability, must have survival value. From logic evolved math which too has survival advantage but it's not necessary that ALL our mathematical knowledge have direct advantage. Some mathematics is pure abstraction I believe but these may provide indirect benefit by augmenting our logical/mathematical skills just like how play and games are very important to a child's mental development.
I think my last post here, is getting to your point as well: https://thephilosophyforum.com/discussion/comment/292353
Quoting fdrake
Notice the implicit suggestion that reason can be understood through the perspective of evolutionary naturalism. After all it is only natural to assume that abstract reasoning skills would be advantageous in the Darwinian sense, in that through them 'the clever hominid' could outwit his nimble, but intellectually-challenged, competitors and predators. But I really don't think it stacks up. It's another form of biological reductionism; that everything about h. sapiens is ultimately explicable on the basis of a theory of biological origins. And ultimately, this can only lead to some form of pragmatism or utilitarianism - because it implicitly subordinates culture to biology. (This is the subject of Thomas Nagel's essay Evolutionary Naturalism and the Fear of Religion.)
Whereas through reason and language, humans are able to transcend the biological; we can understand the biological, and on one level are clearly biological organisms, but it's our ability to reason and to explore abstract truths that differentiates us from animals.
Conversely, it is exactly the capacity to reason which is criticised by some naturalist philosophers, precisely because it doesn't fit into the biological and naturalistic outlook.
Quoting Janus
You've lifted that out of its context.
Quoting Janus
So, I'm not saying "that reason can be understood through the perspective of evolutionary naturalism." if that is taken to mean that some kind of final or absolute explanation can be found there. In any case it would be more of an explanation of the origin of reason in sensory perception, than an explanation of some purported relationship it might be thought to have to some imagined ultimate nature of the world in any kind of platonic sense.
So reason, or logic, is itself irreducible, which means it cannot be existentially explained because all explanations both presuppose and utilize it, as others have noted. I was merely trying to tease out the ways in which reason or logic, and language itself, might be thought to have evolved from our experience of a world of differences and similarities, of change and invariance. Without primordial difference and similarity (identity), change and invariance or regularity or recurrence or whatever term you like, the world could not be the world; there would be no intelligibility to begin with and hence no survival.
And that goes for animals as well as humans, of course. Animals' experiences must make sense to them in their own various ways as ours do to us, otherwise how would they function, let alone survive? It has also been demonstrated that a few kinds of animals (I'm not sure just how many) can perform rudimentary counting, so number is perceptible and makes some kind of sense to them. They must, in their own ways, select form the "buzzing, blooming confusion" of sensory input, just as we do in our own ways.
That is essentially psychologism. Arguments for psychologism haven’t really held much weight historically.
I was offering an explanation to the query ''why logic works or fits so well with experience?'' My explanation doesn't involve psychology. It's based on evolutionary theory though.
Quoting Janus
Mostly me. It’s a form of the argument from reason, although I don’t proceed from there to an argument for divinity, which is how it’s usually presented.
Quoting Janus
But surely the point about animal behaviour is that it can mostly be accounted for in terms of stimulus and response. I don’t *think* animals could imagine how things could be different, so it’s not a matter of them ‘making sense’ of the world, but of responding to it effectively. And in fact, a lot of ‘naturalised epistemology’ wants to explain rationality in just those terms, so as to show the basic continuity of animal and human intelligence. Naturalism doesn’t recognise ontological discontinuities.
Sensory stimuli must be modeled by the animal brain just as is the case with humans. It has nothing to do with "being able to imagine how things could be different" but to do with being able to recognize difference and similarity. I see no reason to suppose that humans have not evolved from common ancestors with other animals, notably chimpanzees with whom we have 99% of DNA in common. Why should the brain be any different?
Sounds like the assertion that claiming there is no objective truth is itself making a truth claim.
As I wrote in earlier posts, according to phenomenology since Husserl, you've got it exactly backwards. Existence is irreducible, and logic presupposes it. There are explanations which precede logic, of which logic is just a historical derivative mode , and not a necessary one. Such explanations do not assume the law of non-contradiction. Differences and similarities are not opposites, they are both implied in every meaning. Invariance is not opposed to change, it is the effect of a constructive activity that maintains itself over time as the same differently. In order to be invariant, a meaning has to reflectively turn back on itself so that it can persist as itself. The effect of exposure to context guarantees that this reflexive move exposes any meaning to alteration of sense. Thus invariance is always the invariance of a meaning whose sense begins to drift at the moment of its turn back on itself in reflection. So the illusion is created of pure invariance only because this continual drift of sense of a meaning is subtle enough that most dont notice it. From this inattention to change within identity was born the concept of pure invariance and the law of non-contradiction.
If I have you stare at an object or say a word over and over again , at the end of this exercise you will declare that the object stared at or the word repeated continues to be the same object or the same word throughout the time frame. But what you likely would not have noticed is that the SENSE of the meaning of the object or word wandered very slightly over that period of time. To claim that this is just a subjective effect and can be separate from what we know of real world objects (and ideal conceptual objects) misses the point that our notion of real world objects is derived from subjective experience.
Logic's assumption of invariance and non-contradiction depends on ignoring these facts.
And physics can ignore them not because the aspect of the world it studies functions differently than subjectivity, but for its own convenience and due to its theoretical limitations it uses a vocabulary that masks these facts.
The problem is, this invariance "works" for predictive models and technological problems. The usefulness of the logic is then what matters. Also, there are fields I am sure, that take into account the variance you describe, and put back the subjectivity in the equation, such as quantum mechanics and relativity, etc.
On a more basic level, tribesman probably see patterns in their everyday living and make note of it in their technology across generations. Agriculture became a pattern that started the idea of living with more coordinated effort to draw water for the crops and animals. This kicked off engineering, and the problem-solving that it needed. Then from engineering we can go to pure logic and math by the time of the Greeks. Other civilizations had their own mathematical systems based on astronomical pattern seeking. So it's the seeing of patterns and problem solving, and having a language to index it all that gave logic its natural force perhaps. All of these patterns are from the evolutionary patterns that helped shape our species, which themselves are from other patterns, down to patterns of laws of nature. Its patterns all the way down man. Our species are just pattern recognizers because of our brain's plastic nature of learning and ability to accumulate cultural knowledge.
Ugh, this is starting to look too similar to @apokrisis :meh:
Obituary for Murray Gell-Man, discoverer of the quark.
Ah, OK. ... Does that make me a psychologist, then? :wink: For I believe that maths and logic were created by humans. They were created for a purpose (or for a number of purposes), and they match those purposes so well - like the glove has five fingers - because we built them that way! We wanted a tool to help us manipulate shapes, making it easier for us to build houses and castles. So we didn't create geometry randomly, we did it with significant focus, to achieve a purpose, and we succeeded.
https://plato.stanford.edu/entries/psychologism/
But I still maintain that maths and logic were created, not discovered.
According to psychologism, everything is a psychologistic argument, n'est ce pas? :smile:
Frege tried to prove that math is reducible to logic and invented "formal" logic as a result. Russell and AN Whitehead tried early on to do a similar project. Many would say Russell himself with his initial paradox and later with Godel and his incompleteness theorem blew this project up- that a manageable set of of axioms can explain all of arithmetic. Neo-logicists try to pick up where Frege left off, but with Russell and Godel's contradictions and inconsistencies in mind.
Why might it be important to tie math to logic? Because while mathematics is not a kind of empirical knowledge we get through the senses, scientific knowledge does work this way. But math is not empirical, it can be done without any knowledge of experience of the world "outside" (gathered from the senses). However, math even if doesn't originate in experiential knowledge, is uniquely useful for experience. You need it to understand the theories of empirical sciences.
By reducing math to logic, it grounds math in a very practical way that humans parse the world (logic). Logic seem much more empirically-based even though it may not be purely empirical. It is closer to human experience than pure math is. Thus, we can see the importance of the project Frege was working on in terms of how math works so unreasonably well in the empirical world (it is based on a more empirically usable system of logic).
However my question goes a bit deeper than that. It is asking why logic itself is so useful. Why does logic work so well? I provided an evolutionary approach for logic's efficacy. This approach tried to demonstrate that patterns in nature (metaphysical statement), by way of some emergentism, have created a being that has pattern-recognizing abilities (epistemological statement). My critique of my own argument is explanation of how emergence works. I've always had a problem with emergence, especially in ideas of theory of mind. However, taking away that tricky problem, it is kind of a basic theory. That is to say, patterns by necessity create more patterns and beings that recognize those patterns. First, it was primitive problem-solving and pattern-recognition, but as accumulated knowledge grew over time (by way of the very basic adaptation that humans developed of accumulating cultural knowledge), logic itself becomes more refined and applied to other sets of problems. All the minutia we monger comes from this.
Edit: An interesting addendum would be that creatures have to follow a "logic" of instinctual norms that more-or-less fit ecological setting (or extinction), or be plastic enough in its ability to recognize patterns (not instinct but cultural and other times of more plastic learning) in order to survive. Due to the necessity of patterns, humans couldn't survive any other way since we evolved such a high degree of plasticity. It would be almost a contradiction for there to be an animal with this much plasticity to not have pattern-recognition and problem-solving skills and eventually logical inferences.
I don’t know how you can. If you offer proof you do so via logical truths. If you claim to need no proof you do so by some necessity you view as illogical?
I would not say I created the world when I was born, I was ‘simply’ born and found myself on a voyage of discovery - and distinguished experiential phenomenon due to faculties of logic coherence. I wasn’t born and then decided to invent logical systems in order to comprehend my surrounding that I couldn’t comprehend or have any comprehension of comprehending prior to creating logic. Get it? Can you argue against that? Would be nice if you could in some useful way.
Neither logic nor maths was invented by one person. And neither was invented, I suspect, by someone newly-born, as your text sort of implies (but I doubt you intended that :wink:). We found ourselves in a world we didn't understand, and we gradually crafted tools to help. Gods and religion came first, and other ideas followed. Your text seems to say that we are born with "faculties of logical coherence", just as we are born with legs. I don't think this is so. Otherwise, our paleolithic predecessors would have been logic-users, long, long before the Greeks laid down their foundations for logical thinking.
You also seem to say that, without logic, we could not formulate logic. But when put as I just did, I think the shortcomings of that argument are clear?
For example, I can construct a sequence of cakes from a bunch of ingredients by iterating a recipe in my possession. But I can also obtain an identical sequence of cakes by repeatedly pressing a button on a vending machine. Neither logic nor set theory take care to distinguish these two sets, because the process of construction is viewed as being either irrelevant to, or identical to, the meaning of "existence".
Consequently a hideous "bugfix" called "the axiom of choice" was invented in order to accommodate "non-specified" entities, causing mass confusion and yielding ridiculous implications in failing to treat sets and their construction on a case by case basis.
Notice that when we point to the invariances of a language like Boolean logic, we are directing ourselves to an inscribed symbolism, something 'physically' present on a page or built into a device. But as I said concerning the way that our sense of the meaing of a symbol drifts the moment we encounter it and then try to return to it the next moment, the evidence of this isnt in the symbol itself but in what happens moment to moment between ourselves and the symbols, in the space between. Its kind of a catch 22.If you believe in the notion of invariance, you can just point to your logical symbol as evidence of it, and if you dont believe in invariance you will also point to those symbols for evidence, but with this 'in-between' space in mind.
The point isnt that logical symbolization is wrong or doesnt work, but that it works not because it is the manipulation of relations between invariant abstractions, but becasue it instantiates a narrative thematics with enough inferential consistency to make us believe in the invariance of its joints. But its the implicative consistency that makes it work. It not only doesnt need invariance, but belief in this concept holds back what our technologies can do, becasue they aren't designed to pick up on and take advantage of this natural drift in sense. instilling greater creative innovation in our machines will require that we explicitly tap into what we now only implicitly understand in our technological languages.
Invariance only works as well as its limitations allow it to, just as Cartesian philosophy 'works' only as well as its limitations allow. This is like saying that those who believe that truth is 'objective' can cite how wonderfully a non-relativistic approach to science solves problems. They cite the wonders of the hypo-deductive method and the linear progress of the sciences. But Kuhnian approaches to science(there is no objective truth), which believe that science is not a linear progression, and that scientific ideas change via revolutions rather than accumulation can argue that their way of understanding also 'works', but differently, and in a way that provides more options for creative advance of thought. This is because "truth is not objective" doesnt mean objectivity is false, it means the idea of objectivity is an island floating on a moving sea, but its adherents cant see past the edge of the island and so see only invaniance. Technologies used to build computing machines work wonderfully, but in a limited fashion.. In order to exceed these limits and accomplish what even the simplest one celled organisms are capable of in terms of intelligence , they will have to modify their vocabulary and methods.
Quantum mechanics and relativity dont question the fundamental basis of an objective causal logic, although they play around with applications of it in terms of specific mathematical models
Conway's game of life was small step in the direction i have in mind.
Suppose there were a way of looking at mathematics and logic that did not involve the issues you are addressing.
So for example you ask why it might be important to tie maths to logic. To do so requires that you treat maths and logic as if they are distinct. But if maths and logic are much the same thing, it would not make sense to seek to tie them together.
Another example is the puzzles around maths not being based on experience and yet being about things we experience. Perhaps there is something amiss in such an approach. As if one could learn to count without there being anything to count. We count the blocks, then the shapes, then the cats and fingers, doing something that is both the same and yet entirely different each time. How could counting not be something we do within the world? Would you express surprise that when you put one thing with another, there are two things? SO why should you be surprised that other sorts of mathematics are useful?
SO, again, it seems that it is your picture of mathematics that is the source of your disquiet.
On your other suggestion, the theory of evolution is itself a broad expression of mathematical and logical notions; so there is a sense in which seeing an account of logic and mathematics in evolutionary terms is still seeing logic and mathematics in logical and mathematical terms. It's not clear that any philosophical progress has been made here.
Also, if you have something to say regarding philosophical relativism and/or psychologism it would help me too.
Thanks
And this. Why must logic and maths be either discovered, or invented. Why not both?
For me this doesn’t seem to have been approached in this thread.
logic is entirely based on definitions and drawing 1 dimension, 2 dimensional, 3 dimensional, 4 dimensional and any higher dimensional relationships between 2 or more defiinitions.
The definition of 5 is (according to the dictionary) is 4 + 1. To know the definition of 5 you must know what the defintion of 4 is.
Does that help you understand?
Remember, we usually need to verify our proofs via appealing to external facts, e.g. a calculator.
A calculator is based off of human and actual logic. If humans can't come up with facts that are rational then neither can a calculator or computer.
At some point everyone needs to agree on basic facts. The issue quite often is some problems that need to be solved need to be solved quickly. Most problems if given the right amount of time can be solved with an optimal solution. A SWAT officer doesn't always do the optimal thing because he/she has to make quick decisions. A judge in a court case has alot more time and has a higher probability of making an optimal decision in a court case.
yes, we appeal to external witnesses, such as the operations of computers, or to the opinions of others, in order to conform whether or not our reasoning is tautologous. But this external checking not only confirms whether or not our reasoning is in accordance with our logical definitions, but it also constitutes part of the very meaning of our logical definitions. For the meaning of "ideal" logic must ultimately be witnessed by practical state of affairs, if it is to have public meaning.
So in my opinion, logical necessity is empirically contingent, although not contingent upon the testimony of any particular external witness.
i'll have to re look up what empirical means but that seems to make sense to me.
Sure that's another way to look at it. I think we have to try to understand why Frege wanted to tie math to logic. I made a "jump" perhaps on why Frege wanted to tie math to a manageable set of axioms and symbolic framework by saying it was due to the idea that logic is more amenable to human reasoning itself.
First let's define logic. Is it purely the forms that were started by the Greeks (notably Aristotle and the Stoics) and later refined by people like Leibniz, Boole, Frege, Russell, Whitehead, Peirce, Tarski, et al? Or is this more "formal logic" of symbols, and relations of the symbols through various methods of inference a specific form of the mere act of inferencing itself?
I think logic as its own genre of inquiry may have culturally started with the Greeks (and other cultural contributors that synthesized with the Greek method etc. etc.), but inferencing itself is simply part of the human animal's capabilities. Our ancestors and tribesman today can inference about a lot of natural phenomena (what causes sickness, what plants have healing properties, etc. etc.) and this inference ability is indeed an informal form of logic. Inferencing from a specific set of circumstances to a broader class (or vice versa), and deducing conclusions from broader notions, are done even at this pre-agricultural level of existence.
Further, being that logic is much to do about observational patterns (a posteriori) and internal patterns (a priori), both forms of knowledge are innately tapped into, by all humans and cultures in some form (though not formalized and distilled into its own genre and then applied afterwards). Rather a rudimentary form was the basis for the distilled/applied form later on crafted and refined by the Greeks through accumulated cultural learning. These, in turn, cannot be helped to be substantiated in the human as it is necessitated by the laws of evolution/survival that animals either follow patterns of instruction (instinct/most other animals), or recognize patterns of instructions (humans).
@fdrake@StreetlightX you may be interested too.
I think we are actually getting at similar conclusions. Look at my previous post about this here: https://thephilosophyforum.com/discussion/comment/292703.
I too think that the kind of logic that the Greeks crafted, and that was then taken up and further elaborated by people like Frege, logicists, and analytics, were refined and "invariant" versions of the more general inferencing power of the human animal in general. It isn't the only version of inferencing, but a very formal version of it. Tribesman's inferencing works in their environment.
Here is where we disagree, perhaps. The more refined version of logic, which stems from the more general inferencing power, has more efficacy in prediction and technological efficacy. Thus, there is perhaps a realism going on that this more refined version is intuiting. Perhaps there are more refined and elaborated logics that are less "invariant" but that doesn't negate the fact that there is some patterns that are being intuited, be them by old-school invariant styles or new-school variant styles.
What is logic as its own distinct identity?
What is logic before we or anybody else imparts any limitations to it?
What is logic before it was a part of philosophy, of mathematics, before it was designated as something to be taught by somebody, etc, etc? (I'm not asking that we should ignore what has been taught over the years but, as they do in science, let us try to track back and see if we could identify logic as itself. It's like investigating the origin of the universe or an earlier state of our earth, let us track logic and observe what it is or could be as an identity.)
We have this thing we call reason or reasoning and most of the time we imply that to be logic. This may be because, as some people say, everything is an illusion taking place in our mental faculties since nothing can be known outside of it. However, the mind is not self/all-existent, is it? We don't consume foods to feed our minds, do we? (And, if we did, how could those foods exist outside the mind, and as what?)
Also, the concept of others (other things, people, subjects, objects, etc) is always in opposition to the concept of our self-identities. If they were a part of ourselves (our minds), there would be no such opposition, would there?
Another thing is we speak of reasoning about this or that, and we know or realise that our reasons and reasoning can be illogical or limited in relation to logic without negating the faculty or capacity to reason. So, what does that mean?
Basically, I'm asking, is there logic beyond our reasons or reasoning and what is it?
Obviously, I'm implying that logic does exist beyond our reasons or reasoning. The identity by which I designate it is what is somewhat controversial, if I may say so myself.
To me, logic is the expression of the laws which govern activity in nature. Here, nature being interactive reality and thus the relative aspect or designation of reality. Therefore, logic designates the mode of operation of nature, whether a specific or collective circumstance, depending on the point(s) in focus. It defines the how but not necessarily the why "things" are.
Which brings us to our application or use of logic. We are limited in our expression of reality. This means we are a part of nature. Also, we are limited in our operation of that nature, which means we are a specific circumstance at best even though we may be a collection of lesser circumstances, fundamentally. Basically, we are not the whole of nature.
We can conjure relations/concepts in our minds which connect to and with every possibility extant in those minds. Such a concept we designate as an absolute. It is a concept which connects to all other concepts, contains all other concepts, exists within all other concepts, influences and determines all other concepts and its identity and character cannot be influenced or determined by any or all of the other concepts. It is itself and as such, an absolute. (We know of such - God, Reality, Life, Energy, etc depending on application.)
But, what is the purpose of an absolute concept? My hypothesis is, absolutes are used to set the limits. In analogy, when we know how high a building should go, we are able to calculate how low the foundation should be for the best stability, and vice-versa. Here, stability being the absolute or the determining factor. I think logic is such a determining factor in relation to our reasoning capacities and faculties.
Does the law of the excluded middle not count here? If not should we immediately stop saying “invent” and “discover” and instead say “disineventcover” or some such term?
This depends purely on the view of what are antonyms and what kind of antonyms they are if they are antonyms. Often people confuse ‘absence’ with ‘opposite’ - I don’t think it is fair to claim that the absence of discovery necessarily means invention in the sense that discovery and non-discovery do. So what kind of an antonym is discover and invent? It seems to me that discovery may come about through exploration or not, whilst invention is a purposeful act (always; regardless of the outcome of the inventive act being useful in areas accidentally). If that is agreed then do we commonly say that something is both purposeful and non-purposeful? Obviously not. So we are not dealing with a gradable antonym it seems. Do we say the existence of discovery requires the existence of invention much like husband necessitates wife?
The connection for me seems to be that we invent, by way of refinement of otherwise, tools to assist in the act of discovery. We do not discover an invention or invent a discovery.
What is discovered can be lost and found again. We could burn all the science books on Earth and stop teaching natural science and logic. Regardless, the same laws and rules of discerning patterns in nature will be discovered in future generations not different ones; meaning these things are discovered yet the means by which we apply ourselves practically to human life means we may invent different pathways - if such a dark age were to happen maybe someone would discover some barely untrodden or neglected paths leading to slower or faster progression toward what we have today.
The only point of contention is in the language used. Do we discover or invent methods of investigation, or is that question badly conceived because it misframes the practical use of ‘invent’ and ‘discover’ needlessly conflating them by using common parse in a seemingly technical manner? I would say that we invent methods and continue to use them if they continue to reveal natural rules/laws to us. We don’t know what we’re going to discover and as we’ve been at this game for a long enough, and consistent period of time, it appears that we are ‘inventing’ when really we’re applying past discoveries toward future outcomes that have degrees of deductive validation regarding what we believe/expect to happen due to causal events - yet we don’t adjust our pure mathematics to fit reality because it is abstract; hence ‘pure’.
A hypothetical pure mathematical deduction can be refuted or proven abstractly. Its relation to actuality is irrelevant to the task at hand.
There is a word like “disineventcover” that indicates the mutual dependence of invention and discovery. It's called 'enaction'. To enact a world as a living organism or not to adapt to an already existing world as Darwim thought, nor is it to evolve independently of ones environment, but to adapt to an environment that the organism is continually reshaping in line with its own self-organizing direction. At the level of thinking, enaction means that we neither simply discover a world that is out there independently of our aims, intents and purposes, nor do we fabricate it out of whole cloth. Instead, we discover a world that derives its intelligibility from our pre-exisitng frames of understanding(worldviews, paradigms).
The problem is, set theory fails to explicitly distinguish 'constructed sets' that correspond to an algorithm known to the logician, from 'discovered sets' encountered externally in the real world, but whose construction is unspecified.
If Set Theory were to insist that all sets can be constructed by an algorithm, then Set theory would also insist that nature is describable by an algorithm, i.e. that a Theory of Everything exists. Yet it cannot ever be known if such a Theory of Everything exists:
Take the example of a vending machine that dispenses a set of items. Should it be the job of set theory to insist that every vending machine has a mechanical implementation, whether or not we know of it's inner workings? Should Set theory automatically assume that every can of coke within the vending machine has a distinct identity before it is dispensed?
Standard non-constructive set theory has a means of specifying an "unspecified set", such as that produced by a mysterious vending machine, namely the Axiom of Choice. But ironically the name of the axiom is a misnomer, because the Axiom of Choice is only useful in mysterious situations where we cannot specify a choice procedure.
Of course existence is irreducible, insofar as it is not an idea, just as anything cannot be reduced to the mere idea of it. The idea of existence or being is also irreducible, it is the most primordial note of logic, but it cannot be explained in terms of any more "fundamental" notion. So. I have no idea what you mean in claiming that "I have it backwards". I also have no idea what you are referring to when you say that there are explanations which precede logic: all explanations are logical, insofar as they have there own logic. If an explanation is illogical, i.e. if it contradicts itself in gross, that is not merely in nuanced, ways, then it simply fails to be an explanation, or even a coherent statement. You don't seem to be making much sense here, but some examples from you may help.
Quoting Joshs
It would help if you addressed what I have actually written. I only quote this whole passage to respond to the litany of irrelevant responses contained in it. I haven't said that differences and similarities are opposites or that invariance is opposed to change. You say it is "the effect of a constructive activity that maintains itself over time as the same differently". It is the constructive activity which is invariant, not the way in which it operates from moment to moment. I have already acknowledged that several times. So, you are just repeating things I have already said in different words and making out as if what I have said disagrees with what you are saying.
I haven't anywhere spoken of "pure invariance", whatever that could even be thought to be. All I can think of at the moment is that it would be absolute stasis, which is impossible, and in regards to which the term 'pure invariance' would not even seem to be necessary and thus appropriate. The idea of pure invariance could only be a purely formal or logical one. Speaking purely logically I could say, for example, that my being myself throughout my life is an example of pure invariance, in the sense that I have not at any moment been anyone else. But of course that is a purely ideal notion, in actuality I am not hermetically sealed off from the world; just in the acts of eating and breathing I am constantly partaking of the world and consequently constantly changing. The invariance consists in the fact that it is I who am changing
Quoting Joshs
What, you don't think I would notice that the thoughts and associations stimulated by staring at the object are constantly changing? :roll:
The point about the logical sense of invariance is that, referring to your little thought experiment, I experience myself as staring at the same object throughout. Without the logical sense of invariance that comes from the apparently completely unchanging nature of at least some sensed objects, which establishes their identity, you would not be able to posit the thought experiment involving staring at the same object for a time in the first place.
The only point about non-contradiction is that you cannot coherently say contradictory things applying to the same object at the same time; for example, you cannot coherently say that an object is simultaneously white and black all over.
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It’s a whole other thread, but I don’t necessarily accept evolutionary accounts of reason. Which is not to say that humans didn't evolve, as we clearly did, along pretty clear (albeit complicated) lines. But when we get to be able to reason and speak, then those abilities really escape the gravity of biology, as it were. (I've been reading about an evolutionary theorist, Kenneth R Miller, whose book The Human Instinct: How We Evolved to Have Reason, Consciousness, and Free Will goes into questions like that. He's not an ID proponent, in fact has testified as an expert witness against ID in US court proceedings.)
I think there's this kind of unthinking assumption that reason evolves, like teeth or tentacles or whatever (to put it crudely) but what see evolving is the capacity to reason - the ability to grasp abstract truths. And I don't think that is accounted for by Darwinian theory as such, as it's not necessarily a question that's strictly biological. (Interesting footnote: neither did Alfred Russel Wallace, who broke from Darwin on this exact question.) So when we perceive necessary truths, etc, we're actually thinking and reasoning in a way that animals generally don't (notwithstanding bee dances, caledonian crows or puzzle-solving octopuses). Hence the Greek definition of 'man as rational animal', which, I think, connotes a genuine ontological distinction.
Quoting schopenhauer1
You're still operating with naturalistic premisses when you say this which, again, you reinforce by re-stating that 'Recognizing patterns becomes the reason why humans can survive'. So, again, this implicitly subordinates reason to survival, (which I *think* is rather similar to what the Frankfurt school criticized as the 'instrumentalisation of reason'.)
We have the picture, or the theory, well attested by evidence, of the ancient universe, before humans evolved, into which we then emerge at a relatively recent time, in geo- and evolutionary terms. That's the modern worldview which is implicitly realist. But what this doesn't see, is that the 'prior truths' that traditional philosophy elucidates, are not themselves a product of this process, but transcend the process. Not 'timeless' in the sense of existing in some 'ghostly ethereal domain' (as we nowadays are inclined to imagine it) but they are foundational to our ability to conceive of time (and therefore develop theories about it) in the first place. So, not temporally, but logically, prior. That is the sense in which traditional philosophy thought them to be nearer the source or ground of being, than what is revealed by sensory perception alone. But what with the abandonment of traditional philosophy, and the concentration on exclusively naturalistic (read: what science can explain) principles, then the original intuition about the significance of the rational intellect proper (nous) has been lost. And one visible consequence of that is the diminishment of the sense of wonder, which, I think, is undermined, whenever we seek to rationalise our abilities in biological terms.
Here's a question that I would genuinely want to get the views from people on PF. Hopefully people understand my question.
Let's assume that there would be an explanation to why we have Russell's paradox and the incompleteness results of Gödel, Turing etc. Hence there would be a central axiom in mathematics, axiom X, that without it we have get into paradoxes and incompleteness results, because we don't take into consideration axiom X, so our logic "breaks down" and we have to settle with ZF-logic or other kinds of logic.
Would there be any other problems with Frege's ideas (naive set theory) and the idea that mathematics is comes out of logic? Is the set-of-all-sets the only problem?
How a human can command a vehicle. Instead of reducing it to each tiny detail, the logic is the simplified event of a human commanding vehicle. It makes sense.
The universe is deeply seated in Logic, when trying to transfer the analogy to the logic behind day, it won't seem fitting because days are not technology, but natural logic.
Sure, weeding out what is an exaptation and what is truly selected for is a tricky area. I am not sure experiments that are/could be done to prove one way or the other. But, we can posit that certainly inferencing ability whether an exaptation that "piggybacked" on an actual selection (better tool-making, increased brain size, better social learning, etc.) did not hurt the animal, and in turn lead to other possible selections that actually refined this ability further. So what once was a "spandrel" (pace Stephen Gould) is now an integral part of the organism. So not only genetic changes, but phenotypic changes that were generally just happenstance, become coopted as a necessary functioning of that organism.
Quoting Wayfarer
I actually agree with you/them here, but for possibly different reasons. We are ontological distinct in the fact that our language ability subsumes everything about our cognition. In order to understand "glass has water" we must have the underlying ability to formulate the world into distinct concepts/use syntax etc. How this occurred is another interesting area that has a lot of current imaginative approaches (see Terrence Deacon's "Symbolic Species" approach for example). Certainly to me, it seems there had to be a synthesis of sexual selection, greater need to pick-up social learning already present in our chimp-like ancestors, and tool-making which was evident early on.
Quoting Wayfarer
Yes that is the point. One can say this is a highly anthropic point of view contra speculative realism. That is to say, that patterns had to be apparent to us in order to survive. These patterns, more-or-less had to be "true" in order to maintain our survival, or we would die out fairly quickly or have to find other modes of survival that do not involve leaps in cognitive inferencing, pattern-recognition, and accumulated knowledge. In turn, contingently, through time we turned that inferencing nature on the world itself and have "hit upon" some fundamental patterns of the world that can be harnessed and used for accurate predictions. These patterns aren't arbitrary, or based on contingent circumstances of culture either. Mathematically-derived empiricism works. If one wants to subsume it in the idea that it is only "useful" that also works, as in this case, what is "useful" is what is exactly what is telling us about the world itself.
Quoting Wayfarer
Perhaps, but there is room in realism with extreme Pythagoreanism (all is math, and we can more-or-less understand some of it, clearly), or Whiteheadian panpsychism, the hyper-chaos theory of Meillassoux, and many other speculative approaches that are "real" in the sense of the theories bieng ontologically grounded rather than focusing on how they epistemically constrained. Of course, I think there is room too for seeing the ontological through the constraints.
Quoting ssu
I'm not sure if he would weigh in, but that might be a great one for @fdrake, though I am not sure how much he is familiar with Russell's Paradox and Godel's Incompleteness Theorem's impact on Frege's logical project. I think Russell's Paradox and Godel's Incompletness is one example of the flaw in the logic itself. There may be broader criticisms that this approach is erroneous to begin with. Math may not be subsumed in a broader logic.
I think that there isn't any flaw in logic. Logic is perfect, we just are not.
The most likely flaw that we have is that we presume natural numbers to be the basis of all math (because from that practical use the field has generated) and also think that we have all the fundamentals of math already at hand. This is our fatal "flaw" here: illogical premises that we are ignorant of.
So we erase the paradox away typically by the axioms of ZF, which however then does contain the axiom of infinity. Then we simply say that what Gödel's imcompleteness theorem refers to has not much if any value. Yet all the incompleteness results do seem to point towards the realm of the uncountable / unprovable yet logical existing. The problem is that people think about this as some kind of attack against math or progress. As one writer called Russell finding the paradox as "a skeleton rattling in the closet far louder than ever before".
We have made such false assumptions before like with the Greeks assuming that all numbers are/have to be rational and were dissappointed to notice that it isn't so.
Getting through the substance of a matter is like decoding logic.
Logic is like mind fodder.
In my earlier post I mentioned, " human driving a car, when functioning, is logic when simplified. Rather than reducing it to the humans cause, it is neutral of cause and effect, as the matter is to be interpreted logically as man moving car, not scientifically but for thought.
We get lost inside our minds when we think using language, when we think about the tools we apply to language and wonder why they fit the world so well, but remove the filter of language and you can see the world clearly again, without the artificial problems we create ourselves in our minds.
This is the view that logic is constitutively normative for thought. That is, the norms themselves make thinking possible, just as the rules of a game are constitutive for that game. They don't "regulate" how the game is played, they are part of what it is for the game to be the game that it is. Without the normative force of the logical laws, thinking is not possible. From the SEP:
Right! Hence my remark in the other thread that numbers (etc) are constitutive of thought. That is exactly what I meant! And as I can see you are learned and competent, I greatly appreciate the opportunity to try and explain this point further.
I agree with the gist of the SEP passage above, where it says that 'the normative force of logic does not merely constrain reasoning, it applies to all thinking.' And I think you could argue that this applies even at the level of language itself. Why? Because language is grounded in abstraction, in judgements that 'this' is 'like that', and 'that' means 'this'. These judgements are likewise constitutive of reason and rational inference, and they are being made whenever we assert or describe or argue anything whatever. They are the 'fabric of reason', so to speak. (For further elaboration on Freger's view of the 'laws of thought' in particular, see Frege on knowing the Third Realm, Tyler Burge.)
But I should take some steps back. The idea that first got me interested in philosophy forums was exactly about the reality of abstract objects (such as, but not only, number.) Now, in the other thread, I said I was dubious about the usage of the term 'objects ' in this context. I think that in effect describing 'concepts' as ‘objects’ is a reification. I accept the usage of the term ‘object’ as a linguistic convention, but I think this usage leads to a basic misunderstanding of the nature of what is being discussed. And the reason for that, is that modern thinking is overwhelmingly oriented towards the 'domain of objects' - the domain presumed fundamental and exclusively real by natural science . That's why in many such debates, the issue of the reality of abstract objects nearly always comes down to the dismissive question 'where could this "ghostly domain" of abstract objects exist?" There's literally no conceptual space for it in modern naturalism, as what is real is regarded as existent, 'out there somewhere', as the saying has it (see the remark on 'animal extroversion' in the quotation below.)
In earlier philosophies - such as scholastic realism and the various forms of Platonism from which it descended - abstract objects are understood to be real in their own right, i.e. to be ontologically distinct from material or phenomenal objects or to possess or indicate another level or kind of reality. (Hence in Platonic epistemology, knowledge of arithmetical forms is categorised as "dianoia" - which appears in modern philosophy mainly in the guise of Galileo's mathematisation of physics.)
One of the profound consequences of the transition to modernity was the 'flattening' of ontology, such that there is said to be only one real substance (ouisia, bearer of attributes), those being the primary objects of the natural sciences specified in mathematical terminology. Western culture is now so steeped in that, that the ability to think in (scholastic) realist terms is forgotten. Whereas, if we admit the reality of conceptual and intellectual "objects", then
[quote=Neil Ormerod]There are a whole range of other realities whose reality we can now affirm: interest rates, mortgages, contracts, vows, national constitutions, penal codes and so on. Where do interest rates "exist"? Not in banks, or financial institutions. Are they real when we cannot touch them or see them? We all spend so much time worrying about them - are we worrying about nothing? In fact, I'm sure we all worry much more about interest rates than about the existence or non-existence of the Higgs boson! Similarly, a contract is not just the piece of paper, but the meaning the paper embodies; likewise a national constitution or a penal code.
Once we break the stranglehold on our thinking by our "animal extroversion", we can affirm the reality of our whole world of human meanings and values, of institutions, nations, finance and law, of human relationships and so on, without the necessity of seeing them as "just" something else lower down the chain of being yet to be determined. [sup] 1 [/sup][/quote]
So that is the drift. It is not exactly what I set out to say when I sat down to write, but I hope it conveys something of what I'm getting at.
Quoting Wayfarer
Typically, we say that rules or, rather, norms are constitutive. If we think that mathematics is logic (logicist position), then we still have the problem of explaining logical objects.
Logic can be constitutive of thought while logical objects exist. That's no problem at all (rather, it presents no more issues that what realists about abstract objects face in any case).
Quoting Wayfarer
Careful quoting Frege's view on this. While Frege likely held the constitutive thesis, he definitely was a realist about mathematical objects :) He would be a great example of holding both that logic's normativity is constitutive of thought, as well as the view that mathematical (logical) objects exist (and not in a metaphorical sense). Also, thanks for the reference. I am familiar with Burge's paper :)
Quoting Wayfarer
I never did that, and the realist position does not endorse this. In fact, Frege wrote a whole paper on distinguishing concepts from objects.
Numbers, for Frege, are not concepts, however. They are objects. If you think that numbers are concepts, then you need to give an account of mathematics in which number terms occurs as predicate terms in the logical language, or as quantificational statements (whether they be first-order or second-order concepts).
The point is that realists think this is not possible, and so numbers have to be full-fledged objects.
What I mean to say here is that the position does not use the term 'object' in a loose, metaphorical sense, or in a sense that "reifies concepts'. It doesn't treat numbers as concepts. We know this because concepts have formal analogues in a precise logical language. Indeed, Frege's insistence that concepts and objects are not the same is reflected in the very syntax of first-order logic (and Frege's logic has a complete first-order fragment).
Quoting Wayfarer
So I think your position is this: we make syntactic distinctions in our formal languages differentiating concepts (predicate expressions) from objects (terms). But the syntactic distinctions do not map onto any metaphysical distinctions between concepts and objects at all.
Ok, but the issue I see is this: empirical objects occur as terms in our formal languages. They are undoubtedly objects. Properties occur as predicate expressions in our formal languages, and properties are undoubtedly concepts (or the referents of concepts, though I take it you mean to use properties and concepts somewhat interchangeable. In any case that doesn't affect the discussion). But then why should terms that refer to abstract objects be taken to be "reification" of what are in fact concepts? Why not take it as evidence that we may have been doing the reverse, i.e., referring to abstract objects as mere concepts, when in fact they were not?
Quoting Wayfarer
Correct: metaphysical naturalism is incompatible with mathematical/logical realism (in the ontological sense). Still, mathematics remains the most significant challenge to naturalism and one which the naturalists have yet to solve.
Quoting Wayfarer
It did, and I hope I have a better sense of your view so that my objections to it may seem more convincing! (or can be more easily dismantled :P)
That being said: I think we should dig out Frege's paper on Concepts and Objects.
Someone would just say that numbers are objects like leprechauns are objects- made up ones. What would it matter if objects are objects if objects can be imaginary? Numbers can be useful, made up objects. So being an object in Frege's own conception would not make something real. It can be useful though.
I think that pragmatism would a good philosophical school. I wonder why Americans aren't so much into it, even if it is genuinely of American origin (Pierce and Dewey).
Quoting Kornelius
I have looked very briefly into that. What I responded to in the Tyler Burge paper was Frege's sense that 'arithmetical primitives' (etc) are self-evidently true, i.e. can't be explained at any lower level, and also the fact that Frege simply assumes this to be the case, and doesn't feel the need to justify it further. This I saw as a residue of Platonism in Frege's philosophy (which Burge acknowledges). Also the fact that the conceptual domain does indeed comprise 'a realm' (specifically, the third realm) i.e. a domain of concepts that really exist, which again is close in meaning to Platonic realism:
[quote=Tyler Burge]...thought content exists independently of thinking "in the same way", Frege said "that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents - "just as a planet, even before anyone saw it, was in interaction with other planets." [/quote]
So, indeed, a 'realist' view - but I think nearer to a scholastic, than a modern scientific, realist!
But please let me explain my approach to the terminology of 'concepts and objects', as I think it is internally consistent, even if it is different to Frege's.
My initial insight about mathematical realism was a sudden realisation about numbers. This was that (1) they do not come into or go out of existence, and (2) they're not composed of constituent parts (although I later came to realise that this last was only true of prime numbers.)
But in this respect, at least, the nature of number is ontologically distinguishable from the nature of phenomenal entities, all of which are limited in time and are composed of constituents. Accordingly, when the mind sees a mathematical or arithmetical truth, it does so in a different manner to seeing an object of sense; it does so apodictically and on the basis of reason alone, rather than mediated by sense - a distinction which is at the basis of what later comes to be called 'hylomorphic dualism'.
When I had that insight into number - an 'aha' moment! - I thought I had grasped that this was why classical philosophy esteemed mathematical knowledge above 'mere sense impression'. And this insight didn't arrive as a consequence of my having studied Platonism or classical philosophy, it was an unexpected epiphany, although in the long period since I haven't found anything to disconfirm the idea and the more I study, the more likely it seems to be true.
So that is why in my particular (and I readily admit, probably idiosyncratic) heuristic, numbers are not 'objects' - because I use the term 'object' to designate 'phenomenal' or 'corporeal objects'. Furthermore, that the domain of phenomenal objects is, properly speaking, 'the realm of existents' or 'the phenomenal domain' - which I also assume to be the domain of the natural sciences.
I have argued this point specifically with reference to an article called The Indispensability Argument in the Philosophy of Mathematics (an article which I find abounds in unintentional irony).
It begins by saying 'Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to debar any knowledge of mathematical objects.' The 'best theory' is described as given by Quine, and comprising the 'abandoning of the goal of a "first philosophy" (i.e. metaphysics). Furthermore, says the article, 'Instead of starting with sense data and reconstructing a world of trees and persons, Quine assumes that ordinary objects exist', which is indeed the basic stance of naturalist philosophy.
Later we read that
But rather than dismissing physicalism on that account, the article then goes on to argue as to why we have to accept the efficacy of mathematics on the grounds of its 'indispensability' - even though its nature seems irreconcilable with 'our best theories'! (Talk about inconvenient truths!)
Whereas, I would rather argue that the capacity for reason is, just as the Greeks said, precisely that which differentiates humans from other animals, and, therefore, difficult to accommodate within philosophical naturalism (which you acknowledge, and which is what makes it highly unfashionable in the current academy.)
Quoting Kornelius
Well, coming back to Frege, that Burge paper says that:
(Emphasis added.) Now, my heuristic around this point is that such things don't actually exist in the sense that 'phenomenal objects' exist. But they are real, nonetheless: hence, real but not existent! It's not that they're non-existent in the sense that unicorns and square circles are non-existent; rather that their reality is purely intelligible or noetic (this is very close in meaning to the word 'noumenal' which is derived from the root 'nous', i.e. 'an object of thought'.)
But this is an ontological distinction which has been forgotten, lost or abandoned in modern thought (although the distinction of 'existence' and 'reality' is still recognised by Peirce.)
That's the sense in which 'intelligible objects' comprise a domain or a realm; but, because of our encultured naturalism, it's impossible for us to conceive of a realm that is not located in time or space. Hence the invariable question 'where is this realm'?
There's a passage in the Cambridge Companion to Augustine on intelligible objects, which says:
Note here again the use of 'object' in the context of 'object of thought'. I suppose my objection comes down to the fact that this suggests that the criterion of the reality of something is it's "objectivity", but that I take "objectivity" to be a naturalistic criterion, as it invariably has to be correlated with something in the empirical domain. Whereas the 'incorporeal' realm is not at all 'objective' in the naturalistic sense. So here we are actually dealing with metaphysics proper - precisely in that sense that Quine, et. al., has rejected. This is where, I think, that Western philosophy lost its connection to metaphysics proper.
Anyway - enough already! Or probably too much.
Quoting schopenhauer1
This person ought to immediately cease and desist from using computers, which only operate by virtue of the fact that numbers are not simply 'made up objects'. It would be more consistent with such a view if they eschewed technology altogether and returned to subsistence farming.
Then they would just say it's the physical output of an electrical circuit opening and closing other circuits. This would be a physical act.
Not all meaning. Linguistic meaning... as it pertains to logic... sure.
Why laughing? I am just saying, Frege seems to think anything is an object as long as it is not a predicate statement. Thus, any old imaginary thing can be an object. That does not seem to be a great definition of an object. In a way, I agree with your interpretation of an object as material.
Well, first of all I was reacting to:
Quoting schopenhauer1
What I tried to get at is that almost everything about the technology we're using every day (or minute!) relies on the 'unreasonable efficacy of mathematics'. We can't "make up" fundamental arithmetic and geometry, or the science behind computers - it is something that has had to have been discovered. Now, I know this then segues into the whole 'is math discovered or invented' conundrum, but I'm of the 'discovered' view - hence my initial (and admittedly sarcastic) response.
But then there was:
Quoting schopenhauer1
Yes, a computer is physically existent, but its physical form is simply the means of interfacing with intellectual or intelligible objects. I argue that logic itself is not a physical phenomenon - it can be expressed or realised or instantiated physically, but in essence, it's the relationship of ideas, not of physical entities. That's why humans can build calculating devices, and Caledonian crows, despite their cleverness, cannot - because humans can 'see reason', so to speak.
Quoting schopenhauer1
No, I don't think he says that at all, but must confess to not having read his 'concepts and objects' paper.
I'm not sure. @Kornelius what would be the difference between numbers and leprechauns in Frege's conception of objects? I realize that question is funny as I write it :).
I think many Americans don't engage in much philosophy at all if you are to characterize it as a whole. I'm guessing that is most societies though, except perhaps France who may put more stock in philosophy as more a political-cultural statement? If you are talking about academics and professors, I honestly don't know what is most popular. My sense is that Pragmatism as well as analytic philosophy in the tradition of Russell, Wittgenstein, Kripke, Quine, Austin, etc. rates pretty high in academic citations.
The populace as a whole I would say values use over other considerations. Technological innovation is a huge part of American industry. However, I suspect use-value is prized in almost all countries. Philosophical abstractions would not rank high as a value for most people anywhere.
Also, being such a pluralistic society, by de facto, Americans assent to a sort of pragmatism whereby if it is useful to you, and it does not interfere with what someone else finds useful, then they all should be considered legally equal. There is a pragmatism with dealing with people from so many backgrounds that have to "get stuff done" in civil society- the everyday aspect of work, consumption, public utilities, etc. I realize however, this is not Pragmatism proper- the philosophy that Peirce described as ""Consider the practical effects of the objects of your conception. Then, your conception of those effects is the whole of your conception of the object."
Overall doing is more important than reflecting I would propose in American values. Reflecting is for idlers and idlets are bad. It's always go go go. If you are doing something, what is its use towards improving your survival, comfort, or entertainment levels would be the implicit assumption. Where are you going? What are you doing? Not what are you thinking, not what is the meaning of. But again, this is probably most societies. Most people dont want to reflect why they should do anything at all or what motivates all this doing in the first place. It assumed an action with some outcome shod just happen.
Hey Wayfarer, I do not have the time for a considered reply to this, but I can reply to the issue about imaginary or fictional objects, so I will do that first and will get back to this.
Quoting schopenhauer1
This is hitting on a point that is actually somewhat contentious and deeply philosophically interesting.
For Frege, the term "leprechaun" is an empty name (or, rather, an empty noun). It does not refer to an object.
The term "three", on the other hand, refers to an object.
But here is the issue. If I say, for example:
(1) "Sherlock Homes is a great detective"
This sentence does have a meaning. But, for Frege (as for many), the name "Sherlock Holmes" is an empty name. But, at the same time, "Pegasus" is also an empty name, and while the sentence:
(2) "Pegasus is a great detective"
Clearly has a different meaning than "Sherlock Holmes is a great detective", the referential semantic components of the two sentences (1) and (2) are the same, given that the names in each do not denote an object, and the rest of the sentence is identical! I say they have different meanings because many would believe (1) to be true, but (2) false, and it wouldn't be because they are mistaken about Sherlock Holmes or Pegasus (or what it means to be a detective).
So how do we make sense of this?
Frege has a differentiated notion of semantic content. That is, terms, sentences, etc., are have both a reference (as part of their semantic content) and a sense (as part of their semantic content). So while the names "Sherlock Holmes" and "Pegasus" don't differ with respect to their reference (they both do not refer to an object), they do differ with respect to their sense.
Now there is a lot of literature on Frege's views on this, and whether he had successfully clarified his notion of sense to properly account for empty names.
But, in short, that is the response. "Leprechaun" behaves semantically differently (so to speak) than number terms since the latter have reference and the former does not.
This is not the only possible view, however. While Frege would not have agreed, many philosophers have since argued for the existence of fictional characters, and thus for the existence of objects that would be the referents of the term "leprechaun". Mostly because it solves a lot of the technical philosophy of language/logic issues (but at the expense of not being consistent with our common sense views about these terms).
Hope this helped!
What is the object referencing? Presumably reference is a "real" thing, but how does he explain this without being self-referential? If he says it is somewhere in the world, then where is this "three"? But if he says it is in the realm of the imagination, then he once again has no way of differentiating it from the leprechaun.
An abstract object. In fact, for Frege it references as extension (of a second-level concept). To be a bit more modern: it would pick out a particular set. Leprechauns are not sets. Sets have properties leprauchauns don't have (and vice versa).
Frege's way of differentiating abstract objects is via definition. For him, a definition must settle all mixed identity claims. So, the definition of the number three would settle whether or not:
leprechaun = 3
is true or false (and it would come out false).
Let's be a bit more modern again, and not use Frege's explicit definition but a semi-analagous one:
[math] 3 :=_{Df} \{\varnothing, \{\varnothing\}, \{\varnothing, \{\varnothing\}\}\}[/math]
It follows from this, for example, that [math]\varnothing\in 3[/math] or that the empty set is an element (or part of) the number 3. But this is not true for leprechauns, i.e., empty sets aren't parts of leprechauns. So they can't be the same thing.
Why can't leprechauns be an abstract object? It may not be a mathematical object, but why not an abstract one? Being a set would be a definition of a mathematical object perhaps, but not all abstract objects are subsumed in that. Economics for example is an abstract object. creativity is an abstract object, etc. How does his definition differentiate between any of these?
They can be, I didn't mean to exclude this possibility; I was only explaining Frege's position on this. But yes, we could allow that fictional entities are real objects of some sort and that the number 3 would not be the same thing as a Leprechaun, because a number is a set, and a leprechaun is a fictional object but genuinely existing object (whatever fictional might mean). It would be an abstract object of some sort and not a physical object.
This would mean that names for fictional characters have a genuine referent. This is definitely a view that is held by many philosophers. You may be interested in the SEP article on fictional entities.