Simple proof there is no infinity
Imagine an empty digital photo, say 800x600 pixels. You could take a camera and potentially go to every single point in the universe and take as many photos from any point in any direction, even using a telescope and microscope, and infrared, ultraviolet, any filter you like… and you can also add to that every frame of every movie ever made, and every page of every book that was written, that will be written, and even those pages that will never be written… also add to that illustrations of every thought and dream, and every scene every man has seen and will ever see....
That single empty photo potentially contains all there is, was, and all that will ever be, and more, even things that can not and will never be. Yet the number of all those possible photos is not infinite. Therefore, if the universe / space is infinite, it can only be due to repetition since the number of unique things that can exist is apparently finite.
That single empty photo potentially contains all there is, was, and all that will ever be, and more, even things that can not and will never be. Yet the number of all those possible photos is not infinite. Therefore, if the universe / space is infinite, it can only be due to repetition since the number of unique things that can exist is apparently finite.
Comments (121)
Quoting NewScientist on physical infinity
Infinity is a principle that arises while reasoning from first principles, such as in mathematics, but not while experimental testing, such as in science. Furthermore, the models for number theory and set theory are never the physical universe. These models are collections of formal language strings. They are 100% abstract only.
Last but not least, you would not be able to prove anything about infinity in the physical universe, because you cannot prove anything at all about the physical universe. We do not have a copy of the Theory of Everything of which the physical universe is a model. Hence, there is no syntactic entailment ("proof") from theoretical axioms possible about the physical universe.
...but chances are humans are kidding themselves in thinking they have an inkling of what actually IS in the REALITY of existence.
We sit here on this tiny rock circling a not especially impressive star in a not especially impressive galaxy...supposing our musings about the REALITY...are of great value.
If all that can be known about what exists were a yardstick...what we humans know may be nothing more than the first atom on that stick.
Fun to muse. We all do it...and probably should be encouraged to do more of it.
But to suppose that we are even close to "proving" anything like "infinity does not exist" or "infinity does exist"...is laughable.
It's simply about the set of all the possible combinations of pixels. It has nothing to do with time and taking photos, that was only to illustrate how enormously huge the number is, and yet it is finite.
There is a significant difference between full color, grayscale, and a picture that can only show black / white pixels. It doesn’t follow ‘there are only two distinct things in the universe’ from the ability to ‘encode all the information in the universe with a digital dataset’. You are obfuscating delimiters between different binary strings encoding many different things. What follows is just that any information can be encoded with a minimal set of only two bits, nothing more follows.
On the other hand, at the core of it all there might be just two things that make up everything, and in some way it is true. Attraction and repulsion for example, plus-minus, things either move apart or come closer together, that is all things really do.
Coincidently, I do have a theory everything is made of only positrons and electrons, but never imagined these two topics would come together, and I do not think there is actually any meaningful connection.
Yeah, that was the point of the reductio. One black/white pixel can encode one bit. 800x600 24-bit pixels can encode 11.5 million bits. But that's just a quantitative difference that doesn't bear on the problem with your argument.
I just proved the number of unique things that can or will ever exist, both physical and imaginary, is finite. What part of the proof can you possibly doubt?
It is not a "proof" as meant by proof theory.
For that purpose, you first need to list the axioms in your theory. Next, you need to show that what you have said syntactically entails from these axioms.
Therefore, it is a problem with (the lack of) formalisms that you use.
I’m not sure if you are saying there is a problem with my argument or not. If there is, point to which statement of mine is supposed to be unwarranted.
You are mistaking mathematics for a self-evident logical fact. In other words, the whole proof is an axiom, what part can you possibly doubt?
Doesn't it bother you that the number of distinct things in the universe is limited by an arbitrarily chosen resolution of your camera? With an 800x600 24-bit pixel camera you can register at most ~8*10[sup]12[/sup] distinct things. If instead you used an 800x600 16-bit pixel you would register at most ~3*10[sup]10[/sup] distinct things. And with a 1x1 1-bit pixel camera you would register only 2 distinct things. Haw can this be?
Hi Zelebg!
Have you considered the concept of time and eternity from Gödel (self-reference), and Turing? Or even the paradox of time itself?
It bothers me. However, whatever new information that mystery holds, I do not see how it can possibly disprove what we can conclude with some standard monitor resolution, but then again, it does feel very strange, so let us think about it...
It does not seem the resolution sets the limit, it’s like the limit is already set pretty low and we only stumble over the limit at some specific resolution. Consider your monitor and Google Earth, you can zoom in and out, so you can encode all the information in sequential images to represent something that is much bigger than the screen.
So we make monitor resolution smaller and smaller, and at what point Google Earth can no longer represent the Earth, if there is such a point? But surely an image with only one pixel and just two colors is meaningless to us, and here we are in the domain of binary encoding, so when and how did we get here, where have we been before, where was the critical point where it broke, and what does it all mean… I’m not quite sure yet. What do you make of it?
I'm not sure. If those are anything like Zeno's paradoxes, then yes, and that does seem related from the perspective of spatial continuity - analog vs. digital "space grid".
Could you please summarize what you have in mind as "the concept of time and eternity from Gödel (self-reference), and Turing," as well as "the paradox of time itself"? Thanks in advance.
Sure, no problem. Gödel and Turing taught us about mathematics being incomplete & incomputable, and never ending, similar to irrational numbers and the modal logic paradox. For instance in modal logic (or if you prefer the liar's paradox/self-reference):
Socrates: What Plato is about to say is false
Plato: Socrates has just spoken truly
With respect to time, another paradox of trying to define time also appears incomplete in its 'eternal way' of trying to measure same. You may have seen this before:
I still do not see what those mathematical results have to do with "the concept of time and eternity."
Quoting 3017amen
Sorry, I do not understand this sentence.
Quoting 3017amen
Thanks for the video. To me, the paradoxes identified by Aristotle and McTaggart are resolved by recognizing that the principle of excluded middle only applies to individuals, while time is continuous and therefore general. There are no discrete instants within time itself, they are artificial creations that we impose for the purpose of marking and measuring time. The present moment is always indefinite, blending seamlessly into the immediately past and future moments, yet itself neither past nor future.
Eternity equals no resolution.
I explained earlier what the definition of abstract time is viz the concept of eternity.
Is that then, abstract and/or eternal?
Eternity usually means timelessness, or perhaps infinite time. I still see no connection with incompleteness and incomputability.
Quoting 3017amen
Not in this thread, as far as I can tell.
Quoting 3017amen
What do you mean by "abstract" in this context? The present is certainly not eternal.
Correct. Otherwise, explain eternity to us?
Quoting aletheist
Time itself.. Can you explain time in better terms? Maybe in less abstract terms, your choice.
Otherwise , it's ok, since you could not grasp my analogy (which is often used in theoretical physics), perhaps you have one that makes better sense?
(If you don't mind, maybe re-post it in different terms with the appropriate analogies so we can all understand.)
Tick toc tick toc LOL
I am attempting to spell out my Peirce-inspired ideas about time in another thread, so I suggest that we continue this conversation over there.
Well, first it is unclear what it is that you claim to have proven. The question you claim to answer is "How many distinct things can there be in the world?" But the answer will depend not only on how the world is, but also on how you cut it at its joints, so to speak: what kind of things are you looking for? Distinct geometrical shapes? Distinct species of animals? Distinct entities posited by fundamental physics? The answer to your question will depend on the chosen mereology, and here the problem is that a priori, without knowing anything about the world, we can hardly even decide on an appropriate mereology. And even if we do know something about the world, our idea of what the proper mereology ought to be can change as we learn more about it. On the other hand, with a trivial choice of mereology (e.g. "Everything is either Donald Trump or not Donald Trump") we can get a pretty good answer without doing any work at all.
But the most glaring problem with your approach is that the answer to your inquiry is completely depended on your instrument of choice. If your instrument cannot register something, then that thing does not exist. If it cannot distinguish between two things, then they are the same thing. The result therefore is bogus: it tells you nothing about how the world really is, it just tells you about the limitations of your method.
Faulty hypothesis. :yawn:
Well, no, the number of distinct digital photos of a given resolution is finite. But so what? He might as well have said: I am going to count the number of distinct things in the world using the fingers on my right hand. Let's see... Tree, house, shoe, smartphone... oops! Only one finger left - I better lump everything else into one remaining thing. There! The number of distinct things in the world is five!
Of course. The entire process is faulty. The assumption that every aspect of the universe can be so pixeled assumes his conclusion.
You don't get it, you can zoom in as much as you wish in arbitrary small steps. You can also forget photographs and imagine all the knowledge there is about everything that will ever be is simply written in English words, with illustrations and diagrams.
You are not addressing the problem. What part of the universe you could not potentialy see on your monitor? You can either name what kind of object or information it is that your monitor can not visually convey, or you have to admit your monitor can convey any and every possible information.
In other words, there is no actual infinity of discrete objects; but this does not rule out real continuity in the universe, such as that of time and space, which are not composed of distinct parts.
A pixellation of something is a map of it to a finite set of polygons that cover it [hide=*](with properties on the polygons that represent colours) (and do not exceed its bounds when scaled to the photo size) (and probably other constraints like the pixels forming a grid on the original object)[/hide]. Let's say they're squares. It doesn't matter what you apply the pixellation to, it ends up finite. You can throw squares on the unit square [math][0,1]\times [0,1][/math] by dividing it into 4 squares along its non-diagonal lines of symmetry, but the unit square is uncountably infinite. That is, an object having a finite pixellation is not sufficient for it being a finite object (specifically, of finite cardinality).
I think you're confusing the necessary finiteness of the pixellation with the finiteness of the pixellated object.
Do you know what kind of properties a space would need to have so that every subset of it could be covered by a finite set of polygons?
Wow! So there are an infinite number of pixels in each photo. You're correct. Guess I don't get it.
Quoting Zelebg
There are a finite number of existing English words, but if there is no limit on the length of a word you could be speaking of infinities. There are 26 letters, so how many "words" could be constructed, of any length? Of two letters: 676. It goes up from there. 26^n. n increases without bound. Also, there could be an infinite number of diagrams.
Is the map the territory?
You are on the right track now, just wrong side of the equation. The number of pixels is finite and so is the number of their combinations, that part of the equation is known. But for the equation to be equal the number on the other side must be finite too, and that number represents every possible information your monitor can represent. Therefore, the total number of unique bits of information is finite, or there is some kind of information your monitor can not display, for some reason.
Can you explain that by describing a type of object or information that can not be visually represented on a computer monitor?
Yes.
A pixellation consists of a finite number of finite pixels arranged in some sort of grid that cover an object.
So I took an object which has infinitely many things in it, all the points contained in a square whose side lengths are 1 and gave it a finite pixellation, four squares defined by its non-diagonal lines of symmetry.
Such a square has infinitely many points, but it nevertheless has a finite pixellation. So an object having a finite pixellation doesn't prove that the object is finite.
What information is lost? Imagine that you're trying to specify your position on the monitor's image - you can represent this as the position of a pixel (its centre). If you move right one pixel, you move a distance according to the length of the pixel. You can't specify any within pixel coordinate just using pixels.
If instead you want to be able to move right by any distance at all, you're already dealing with an infinite object (at least something like the rational numbers, the fractions, which have a coordinate between every pair of coordinates, no matter how close they are).
Say each piece of information is a string of alphabet symbols. Since the length of these strings is unbounded, so is the amount of information. We're not talking about computer programs that terminate.
Quoting fdrake
Compactness? Are your "polygons" abstract entities? Topological spaces or what? Compactness in TS if you adjoin limit points, I suppose. :nerd:
It’s not about size. Infinitely large square we can scale down to arbitrary small size without omitting any information, or even fully describe it just by a single word. It’s about unique features, so ultimately it is about compressibility and randomness.
Forget the images, let’s just take black&white monitor, just two colors and only English words, symbols and numbers from ASCII set. Is there any part of the universe, any law, property, force, event or phenomena, any planet, star, or galaxy, that can not be fully and extensively described on such a monochrome monitor with just ASCII?
For the amusement take a note with the above example we used only a tiny portion of all the available potential space. There are still many empty screens waiting that can hold all that information written in every other language, current, past and future, also all the alien languages included, plus much, much, much more unused space waiting and we have already described every possible thing many times over.
This was my thought too. If covers need not have a finite subcover, then something like a pixellation couldn't exist for sets that fail to have finite subcovers.
My other intuition was that: open covers having finite open subcovers in some topology (of a suitable object) would probably let you "push" any open set in that topology into an open set in the plane through a continuous injective function, then you could cover the open set in the plane with a grid of polygons; composing the continuous injection with the point->polygon grid assignment would give an association between the points of any open set in the first object with a pixel, then you 'pull' the point back through the composition. I didn't check if this preserves the grid like properties on the first space (maybe small open sets in the first space can be guaranteed to hit multiple pixels).
Quoting jgill
I was imagining closed plane figures with straight line edges.
Quoting jgill
Yes!
Quoting jgill
What is TS?
On every page there is a description of a single particle, where it is, what is doing at the given time. Collectively all that information describes everything that exists and will ever exist.
The question is whether this encyclopedia of everything has infinite number of pages or not. The answer is no, because there is no reason why your monitor could not display any of those pages, and the number of pages your monitor can display is finite.
.
Topological space.
Quoting Zelebg
Let's say particle alpha is under consideration. We measure time in seconds. Page 1, present time. Page 2 , 1 second from now. Page three, 2 seconds from now, etc. Page N, N-1 seconds from now. You would have to assume time stops at some point in the future in order to secure your "proof." So you would postulate that time is finite. But this seems to be part of what you wish to prove.
I must be missing some important debate points here, in my old age. :gasp:
Maybe time just goes in circles? It's not my goal to prove anything, it's all the same to me. I just like mysteries and here is some mystery it's not even quite clear what the mystery actually is. Mysterious mystery is the best mystery of all.
If your monitor - or, say, any device or method for identifying distinct objects - can only register a limited number of objects, due to the way in which it is constructed, and you have registered that many objects, then all that you can say is that there exist at least that many distinct objects. This is the point that you fail to grasp.
Suppose, for example, that your device can register only up to five distinct things, and suppose that the world has more than five distinct things. What conclusion do you draw from this: too bad for your device or too bad for the world?
This idea, that your device or method can bias, limit or even fully determine what what you can observe is known as observation selection bias or observation selection effect.
Quoting Zelebg
Perhaps you will realize your mistake if you reduce the size of the page to the extreme (although a similar exercise with reducing the number of pixels on the monitor failed to convince you). If you only have one character on the page, and there are, say, 100 letters, digits and other signs that you can depict with one character, does this mean that there cannot be more than 100 distinct entities in the world?
You keep avoiding the question. If my monitor is limited to show only some of all the possible objects, what is it about those remaining objects that prevents my monitor from showing information about them, why can it show object A but not object Z?
Does it not bother you every time instead of addressing the question directly you always make up your own interpretation and end up answering your own question instead of mine? -- What is the reason why my monitor could not display any of those pages from the encyclopedia of everything?
Ugh, why do I even waste my time on this...
I’m repeating the question so you don’t fool yourself that you have answered it.
No. You claimed there are objects that my monitor can not represent, I’m asking what is it about those objects that prevents my monitor from showing information about them, why can it show object A but not object Z?
I'm not saying your wrong, but something to note when you change the space between two particles or two eye balls you also change the appearance and also the behaviors of that object/ball of mass/or human face. The structures of atoms is effected by distances (as well as other things) between any given "sub" particles or sub atomic particles that make it up. So once again i'm not saying you are wrong however when you change a pattern even slightly or change distances even slightly you are also changing alot of other things. So it is possible for an infinte different variations or patterns.
You are clearly a glutton for punishment!
You can bring a horticulture, but you can't make him think.
|>ouglas
You can't fit all the required information about even a single particle on any finite-sized pixelated page because some of the values associated with the particles would be represented by Real numbers not Integers or Rationals, and a Real number contain an infinite amount of information in it. I.e., you cannot encode an infinite amount of information on a page that can only hold a finite amount of information.
|>ouglas
That is not the answer, just refusal to accept the premise of the question, and is beside the point since the bottom resolution can be fixed to arbitrary size and precision. Say, human faces. My monitor can show every possible human face at least down to a scale and precision of an electron microscope. Therefore, there is only a finite number of unique human faces. Yes?
Simple proof that your theory is false:
Let's assume that the space is infinite and physical manifestation is limited to a finite (not infinite) number of possible arrangement.
Let 's further assume, that A represents the number of possible configurations, B represents any configuration, and C represents a subconfiguration to B.
In this case all you have to do is add C to B, and you got an additional number, A+1, to represent the possible configurations of matter. Obviously A+1 is a number greater than A, and B union C is a unique, new representation.
Now let's assume A' is A+1 and B' is B union C. Repeat the experience, and you realize this experience can be repeated without any ending, and with forever producing A' from A and getting a larger number every time, and with forever producing B' from B without repeating the same configuration.
B here is a given configuration of matter, C is a subconfiguration of B, and BuC means B union C, because u means union (as in set theory) and n is any positive integer greater than 1.
The question is, can my monitor represent information about every possible object or it can not. What is your answer?
But that's not my point. Even with a finite number of combinations, you can present double the amount of combinations if you add another screen or monitor. And triple it with adding a third screen. ETC.
To give a straight answer, no, it cannot, if the information is to be complete, exhaustive and precise.
However, information may mean "limited but pertinent knowledge" or it can also mean "scanty knowledge".
Your may want to rephrase your qestion?
In one single screen, or in a series of possibly different screen shots, of the same every possible object?
The more you examine it, the more your question seems to be watered down which can't be by good faith answered with a simple yes or no, without further clarifications established.
Douglas Alan raised a valid point: you are talking about the visual, he is talking about the real. You asked if anything can be depicted; yes, but not everything can be represented. And your initial premise was that infinity is thus denied. But it is not denied, only the depiction of the infinity is denied.
And that has already been established. How can you make a picture of something infinite? You can't. The picture is by definition a limited, finite area.
Thus, your claim that you can't take a picture of infinity, or can't view it on the screen, is true, but it does not deny the fact that things can be infinite.
It would be analogous to a thought. You can't see a thought, you can't depict it, but you can't deny its existence.
You made an argument with a false premise. Consequently, you have not proven your conclusion. This is Logic 101.
No. Faces can differ in details that are smaller than the resolution that can be captured with an electron microscope. Also, different faces, even if they look the same in a particular pair of photographs, can move very differently from each other, which can completely alter our perceptions of what those faces look like.
|>ouglas
Can any information be digitally encoded? The answer is yes, and to arbitrary given precision. My monitor can indeed represent any and every possible information.
And I am not arguing what you are refuting. I want to set up a basic unambiguous premise we can all agree with and thus have some starting point.
So, for some arbitrary given resolution and some arbitrary given size of an object, such that it maximally occupies the whole screen, say 800x600 resolution and passport style photographs of human faces - there exist a finite number of possible human faces for that particular specified size and resolution. Yes?
No. The monitor has 16 million colours in 1280 times 720 pixels. That gives you a combination of 1.6*1.28*7.2*10^12 combinations. That is not infinity. It is even smaller than the number of atoms in a human body.
Your computer screen can represent any and every possible information up to a combined total of about 10^13 combinations. That is no more than ten thousand billions. That's the maximum number of uniquely different representations that a computer screen can provide
And it proves nothing, actually, of the infiniteness of the combinations possible.
Please tell me why infinity is disproven if your computer screen can represent 10^13 combinations. There has to be a logical link between the two, otherwise the proof is spoof.
Right now you have failed to provide that logical link.
Well, as soon as you reduce the precision to below 100%, you lose information. You retain and pass SOME information, but not ALL information. That is the limitation of your computer screen.
It has ten thousand billion pieces of distinct pieces of information.
But a Kg (about two lbs) weight of ANY gas has over 10^23 atoms in it. Each atom is moving in a different direction, at different velocities, at different spins.
How can you even imagine that with a loss of ten billion TIMES the more information just on the NUMBER OF PIECES of atoms you can pass down any precise information?
It's like taking a human body, and taking one millionth of a millionth of its weight, and declare that you passed the information on that human body perfectly.
But that just proves that your computer screen does nothing of a true or even approximate representation of any complex object.
However, you STILL have the task on hand, to show to us, your captive readers, how this by now infamously poor representation proves that infinity is impossible.
Actually, the answer is no. There are analogue quantities, that can't be digitized. 1/3, for instance, is impossible for a binary computer to digitize. And it will lose some information if it tries.
In triary computers, yes, 1/3 could be digitized, but 1/2 could not. You can't escape this problem with any digital system.
So your observation and stance that any information can be digitized, is totally wrong.
When do you stop being wrong, @Zelebg? Now, there is an irrefutable instance of inifinity for you.
Sure, but so what? Nothing interesting results from this.
If you want to get to the interesting question, let's take Max Tegmark's argument that in our Hubble Sphere, there are only a finite number of possible states. (Our Hubble Sphere is the area of space that is causally connected to us. I.e., it's radius is defined by the farthest distance from us from which light from the Big Bang has reached us.)
If the universe is flat, then it contains an infinite number of Hubble Spheres, and consequently, if you were to be able to travel at faster than the speed of light, and you went far enough, you would eventually come to a Hubble Sphere that is in the same state as ours. Consequently, this is a way in which there might be parallel "universes" that are identical or very similar to ours.
This argument rests on the premise that all the physical features of the world are quantized, however. And this may or may not be the case. If it is the case, then Tegmark would seem to be correct. If it is not the case, then his argument fails because it is based on a false premise.
|>ouglas
I certainly don't agree that with Zelebg, but this assertion of yours is wrong. Computers can and do represent rational numbers at times with perfect accuracy. This is done by representing them as a pair of integers, rather than in a "floating point" format.
|>ouglas
Now this is funny. Don't you see that is exactly what I'm saying? All I have to do is set my arbitrary resolution to planck scale and define the arbitrary given size as that of the universe to match Tegmark.
I hear what you are saying. But the emphasis is on, what you described as, AT TIMES. That is, not always.
Once you enter into a variable the value of 1/7, and you use that variable's value in calculations, you will immediately lose the perfect accuracy, as the calculations storage go on binary code representation.
Yes, I have agreed as much. The problem is that Tegmark is making a contentious premise in his argument, and therefore, we cannot be sure of his conclusion. All we can say is that if his premises are right, then his conclusion seems to be right, but if his premises are wrong, then we can have no confidence in his conclusion.
|>ouglas
This is not true. A programming language that supports doing mathematical calculations with rational numbers will typically not force you to ever convert the rational number to a floating point number. The program can run from beginning to end using only rational numbers, and can consequently produce results with perfect precision and accuracy. (Assuming that the numbers being represented are accurately represented as rationals.)
|>ouglas
P.S. Here is an example of a library for Python that lets you do just this:
https://www.tutorialspoint.com/python-rational-numbers-fractions
Perfectly true. But the numbers will be thus represented as long as a program is run written in that particular programming language. If you run a different program, written in a more conventional programming language, that does not have that feature programmed into its structure, then you lose accuracy of rationals with infinite repetitions.
This argument does not invalidate mine, where I pointed out your reservation, "AT TIMES".
Maybe in the future all programs will run that way. But not at present.
I don't understand your argument. We should not be making any metaphysical conclusions based on how computers are typically used today.
As for the "conventionality" of programming languages, all of the most popular programming languages in use these days have libraries for doing math with rational numbers (and never having to convert them to floating point). These languages include Python, Java, C++, etc.
|>ouglas
A fringe benefit for sure!
|>ouglas
Bring them to the same denominator? Like humans?
I've been out of programming for 30 years now. You are talking, to a real, live dinosaur, |>. It's exciting, innit? Until I devour you in two bites.
Precisely so!
|>ouglas
No, there are algorithms to determine the greatest common divisor and least common multiple of two Natural numbers.
|>ouglas
Right. So if we don’t make any assumptions and instead choose arbitrary resolution and size we can make conclusions related to that specific resolution and size, like: there is only finite number of planets that look unique as seen from the altitude where they maximally occupy the given screen area.
Then we do smaller, lakes and mountains, then plants and animals, and everything else. And while you can argue there can always be some difference further you zoom in down below the decimal point, once we pass the size of an atom those differences are insignificant compared to the more general point.
Ah I see now! You are smarter than the even the greatest minds of our generation. Forgive me for ever having doubting you.
|>ouglas
Okay, if there is a fixed computer we're using.
But if the photo is say K x L pixels, and each pixel contains N bits of information, then by increasing N (to represent hypothetical better and better computers) then the number of K x L photos that can be conceived is infinite.
I haven't followed the discussion but all questions as to the existence of mathematical objects come down to the question as what you regard as mathematical existence. Most people accept that the are "abstract objects," a phrase with a SEP page, and that abstractions live somewhere other than in the physical world. Of course then you have to explain where exactly they live. It turns out to be the same as how you judge the existence of Captain Ahab. In fact if you just regard math as pure fiction no different than a character ina novel; you save yourself a lot of trouble, philosophically. This idea is called mathematical fictionalism.
So whether the number 3 exists or whether [math]\aleph_{47}[/math] exists are the same question. They either both do or neither do; because their existence is demonstrable in standard set theory and agreed to by the world's mathematicians. Everything after that is somebody's value judgment.
I have recently come to a definition of mathematical existence. A thing has mathematical existence when a preponderance of working professional mathematicians say it does. In other words the meaning of the word existence is in the way we use it.
There was once a lot of opposition to [math]\aleph_{47}[/math] but people got over it and today we teach it to the undergrads and explain it on Wiki pages. The number went from mathematical nonexistence to existence by virtue of people getting used to Cantor's brilliant revolutionary ideas. A revolution that brought a new class of things into mathematical existence: the rigorous theory of the transfinite ordinals and cardinals. Their eventual mainstream acceptance brought them into existence.
Of course @Metaphysician Undercover would (and already has) pointed out that I have said nothing at all since whatever mathematical existence is, it can not possibly be something that is historically contingent. But it is. What we call numbers today, like negative numbers and complex numbers were regarded with horror and opposition from the mathematical establishments of their day.
So: If mathematicians say something has mathematical existence, then it does. There is surely no objective standard. It's a mistake to believe that there is.
I would add that Descartes, when he did math, want to see the whole series of proof within one "vision of intuition". Deductive logic will always say there can be eternal contradictions, but if you find your vision of intuition and make it fluid, your mind will be like water too and your Zen masters will be proud of you. This is what Hegel did. He said to find your infinity primarily in the infinity of the world. You've never lived in Plato's cave
So are you saying that when Georg Cantor first defined infinite sets ca. 1871 and there was great resistance among the world's mathematicians, infinity didn't exist yet?
From our past discussions about this, I understand the underlying sensibility here, but I think that it goes too far toward the subjective. Again, I endorse Charles Peirce's definition, which he adopted from his father Benjamin: "Mathematics is the study of what is true of hypothetical states of things." For me, mathematical existence is shorthand for logical possibility in accordance with an established set of definitions and axioms. Mathematicians may not yet recognize something as following necessarily from them, so it is not a matter of whether they do say that it has mathematical existence, but whether they would say that it has mathematical existence upon discovering a proof.
Have you considered irrational numbers, imaginary numbers, and infinite electrical resistance (as measured by ? on a simple ohm meter)?
Well put.
Yes that's what I'm saying. Because the opposite of that proposition is the claim that transfinite set theory was "out there" somewhere waiting to be discovered. But if that's true, where was it? When Ogg the caveperson first made marks in the dirt to count the mastodons killed by the tribe, did all of modern transfinite set theory already exist? Where, exactly? What else exists there? God? The Baby Jesus? The Flying Spaghetti monsters? Platonism is just as hard to defend.
Did Captain Ahab exist before Melville wrote him into existence? Perhaps you can answer me that.
Of course I understand your point, that mathematical objects often seem inevitable after we discover them. But math is a social process. When the preponderance or consensus of working mathematicians accepts a new idea, that social process is what brings that idea into existence. Or at best, when it's first published.
You could ask if the sculpture is there in the stone before the sculptor gets to work.
I understand that we could never get to the bottom of these questions. I'm trying to cut through the existence arguments that are being made here. I don't think you can point to mathematical objects and say that there is an ultimate or absolutely true answer to whether they exist. People argue about whether certain mathematical objects exist or not. Your question is a good one. Where were the transfinite numbers before Cantor discovered them? I have no idea. If you don't think Cantor brought them into existence in an act of human creation; then where exactly do you think they lived? Were they created as part of the Big Bang? That idea is not tenable.
Ok, good point. When someone first makes a discovery, that's when the existence happens. Social acceptance decides the importance.
But still, where was the discovery before it was discovered? As much as it's problematic to claim it had no existence beforehand, it's just as problematic to say it did. If it existed before you discovered it, did it exist at the moment of the Big Bang? How did that happen?
Platonism's harder to defend than fictionalism.
Better to say it didn't exist before you thought of it, just as Captain Ahab didn't exist before Melville thought of him. If we can figure out where Captain Ahab lived before Melville created him, then we can talk about whether [math]\aleph_{47}[/math] existed before the Big Bang.
Quoting aletheist
Not subjective, social. My own hallucinations don't exist. But if I can convince enough people to believe in them, they do. But as @jgill pointed out, a mathematical discovery comes into being at the moment of discovery.
But what if the moment of discovery turns out to be a mathematical error? Then the community corrects it. There's a huge brouhaha in the math world going on right now in the [url=https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/[/url] Mochizuki's claimed proof of the abc conjecture. One group of mathematicians firmly believes a certain result has been proven; others firmly disagree. We have to wait to see how this is ultimately decided, perhaps a long time.
Quoting aletheist
I think that's limiting. It puts trivial conclusions derived from meaningless axioms on the same level as the deepest results. Math isn't just cranking out theorems from axioms. It's cranking out theorems about mathematical objects. No number theorist believes that Fermat's last theorem is merely a theorem that falls out of the axioms of set theory. Wiles proved that FLT is true. True in a way that transcends axioms. It's a truth about the natural numbers; not merely a truth about proofs in a formal system. That's the Platonist in me speaking.
Quoting aletheist
Yes ok. Mathematicians tend to believe they're studying mathematical objects, though; and not just searching for proofs. I think that when you do math, you tend to be a Platonist; but when you try to defend the activity rationally, you have to fall back on being a fictionlist.
As I see it, mathematical truth exists independent of whether there are any conscious beings who know about it.
Quoting fishfry
Not really. Notice that my definition requires the set of definitions and axioms to be established, which could be interpreted as consistent with your requirement for intersubjective agreement among practicing mathematicians. The "deepest results" come about when someone works something out that follows from those definitions and axioms, but either has not been noticed or has not been demonstrated previously.
Quoting fishfry
It's a truth about the natural numbers as established by a certain set of definitions and axioms. The latter are the only way we know what anyone means by "natural numbers."
Quoting fishfry
Quoting Daz
I am more in agreement with @Daz on this, but would substitute "is" for "exists" since the latter has ontological implications that I wish to avoid. Platonism holds that mathematical objects exist in some ideal realm, while fictionalism holds that all properties of mathematical objects are dependent on what someone thinks about them, just like characters in a novel. I am a mathematical realist, but not a platonist; I hold that mathematical objects are real by virtue of having certain properties regardless of what any individual mind or finite group of minds thinks about them, but they do not exist because they do not react with anything. Fermat's last theorem would be a truth about the natural numbers, as established by a certain set of definitions and axioms, even if Fermat never conceived it and Wiles never proved it.
This is by analogy with the intuition that material objects exist. These exist in the sense that they can be imaged, measured and perceived by many people who will get the same measurements of the same things, at least within a fine tolerance that would improve as our measuring instruments improve.
Similarly, mathematical objects and truths can be observed and measured by many people, or even software, and multiple computer programs (assumed to be debugged) and mathematicians (assumed competent) will observe the same truths about these objects.
This perception of material objects, or mathematical truths, doesn't need to occur in order for the things at issue to exist; it just needs to be in principle possible.
Let me add that I believe that if any hypothetical beings, or even high-powered computers, were acting like mathematicians in distant parts of the universe, they would eventually come upon the very same mathematical truths that mathematicians on Earth would, perhaps in a different sequence. Math is like a landscape waiting to be discovered, no matter where the discoverers happen to be.
The fact that you say "every photo, every dream..." does not entail that there is a finite number of photos or dreams.
It seems you have assumed that there is only one thing called "the universe", and that it is finite in space and time.
How do you know this? What is the basis for this claim in your argument?
Depends on what you mean by "the universe". If the nature of the universe is established via the scientific method, whatever is the result must be finite.
It seems to me that it's only what's called "the known universe" that is "established by scientific method".
But there's an important conceptual difference between the world as it is, and the world as it is known by us.
I see no reason to suppose that our knowledge of the world at any given time in history would give us complete knowledge of the whole world.
Is there some reason to suppose that what we know about the universe at any given time, in keeping with scientific method, is all that we will ever come to know?
Is there some reason to suppose that the sum of everything we could ever possibly know about the universe, in keeping with scientific method, would provide a complete account of everything that is in fact the case, across all time and all space, or across whatever "dimensions" we should name alongside or instead of time and space, and across whatever universes and multiverses and iterations of generation and decay of universes or multiverses there may be....?
The known and the empirically knowable, yes. But beyond that, the meaning of "the universe" gets rather vague and nebulous.
Quoting Cabbage Farmer
This topic tends to run into language limitations. So, I get what you're saying, but the problem is that concepts like "knowledge" break down when we go beyond whatever we can somehow experience. Everything we'd "know" about the "world as it is" can only be based on deductions from first principles, something entirely different in nature from knowledge about the physical world.
Based on that, we cannot ever "come to know" anything about the "world as it is". If that information is available to us, we already have it, we merely need to make the correct deductions.
Quoting Cabbage Farmer
Well, yes, because by definition "what is in fact the case" is established by the scientific method. You probably mean that there might be large parts of reality forever hidden from any human mind. And that could be the case. Or it could not. But for practical purposes, it seems irrelevant.
I should be careful. You described the point of view of Peirce, and I criticized that viewpoint. But I don't know enough about Peirce to really comment intelligently. I don't have certainty in my own positions. I'm a formalist when it suits me and a Platonist when it suits me. I don't want to let my rhetoric exceed my actual understanding; so I'll just say that Peirce probably knows a lot more about this than I do. And I haven't ever given any thought to mathematical existence. Mathematical existence is perfectly obvious to me from the perspective of math. It wasn't till @Metaphysician Undercover challenged me to define it that I ended up with my current position. But I'm not very wedded to it, nor is getting the right definition important to me. When I do math, mathematical existence is perfectly obvious. The question never comes up.
So rather than try to defend my position, I'll just say that what I've written so far represents the limited extent of my thinking; and that I'm not going to think much about this anymore. It's perfectly obvious to me that [math]\sqrt 2[/math] exists; and nothing I say will ever convince @Metaphysician Undercover. I should just quit while I'm behind.
Quoting aletheist
Yes, on this you're just wrong. No mathematician thinks that way. Philosophy is not about standing outside a given discipline and telling them they're doing it wrong. Philosophy has to be about explaining what practitioners are actually doing. FLT is a statement about the natural numbers, and everybody knows exactly what they are. You do too; and no invocations of philosophy or nonstandard models will change the fact that you have an intuition of the natural numbers, and that I know you do.
One must account for that in one's philosophy. There, now I'm being a Platonist again.
Quoting aletheist
Where does mathematical truth exist? Did it exist before there were humans? Before the solar system? Before the Big Bang? What else lives there? You can't avoid the ontological implications. Mathematical truths are abstract things, so they have abstract existence. But they're based on things outside, like counting pebbles.
I agree with you that FLT was true before Fermat. Mathematical truths are very strange philosophical things. On the one hand they're nothing more than valid inferences from arbitrary axioms; and on the other hand they are obviously eternal truths. Quite the mystery. Beyond my pay grade I think.
No worries, I always appreciate your point of view on philosophical aspects of mathematical matters.
Quoting fishfry
I am not telling anyone that they are doing something wrong. I suggest that philosophy is (among other things) about explaining what practitioners are actually doing, regardless of whether they accurately recognize it themselves. I have written extensively about philosophy of engineering, my own discipline, and colleagues find it fascinating because they never otherwise think about it in the way that I explain it; they just do it. For better or worse, most practitioners are not reflective practitioners in that sense.
Quoting fishfry
Yes, in accordance with a certain set of definitions and axioms. Since we cannot point at a natural number to indicate what it is, all we have is a hypothesis from which we can and do draw necessary conclusions (like FLT).
Quoting fishfry
In Peircean terms, "abstract existence" is an oxymoron. Some abstractions are real, because they are as they are regardless of what any individual mind or finite group of minds thinks about them; but no abstractions exist, because they do not react with other things in the environment.
Your claim "If the nature of the universe is established via the scientific method, whatever is the result must be finite", seems fair enough if it's a claim about the finitude of the current results of scientific method at any point in history, a claim about our knowledge.
If, by contrast, you mean that "the nature of the universe" is itself identical to the results of scientific method at any point in history, then your claim is indeed the product of vague and nebulous confusions. It seems likely to me that this claim is closer to what you intended, so I'll proceed here on that assumption:
Such a claim would resemble Zelegb's claim to have provided "proof that there is no infinity". Both claims purport to aim beyond what is empirically knowable. At most you can claim to show that our knowledge of the world is finite. But you cannot claim to show -- or how would you show? -- that our knowledge of the world gives us a perfectly complete account of the world as it is in fact.
By my account, those claims of yours and Zelegb's amount to speculation beyond the limits of empirical knowledge, and seem motivated by unwarranted conceptions of the relation of knowledge and reality.
By contrast, I have not claimed that the world is infinite. Rather, I say
(i) it seems we cannot know whether the world is finite or infinite in the relevant sense
(ii) surely the fact that our knowledge of the world is finite, or that "the world as we know it" is finite, is no proof that the world itself is finite
(iii) your claims seems to contradict both (i) and (ii).
You have conflated our knowledge of the world with the world itself, and thus engaged in a sort of metaphysical speculation. My claims are claims about the limits of knowledge. Your claims are claims to know the limits of the world.
Quoting Echarmion
Indeed. It seems to me these limitations are very much at issue here.
Quoting Echarmion
Here again, it seems to me you've let your speech drag your claims and your beliefs beyond the bounds of evidence and reason.
Don't you agree that what is in the fact the case is in fact the case, whether or not we know it? Or do you suppose our knowledge creates reality in every regard?
Our knowledge of what is in fact the case is informed by experience and is made rigorous by scientific method. That does not entail that experience and scientific method establish what is the case and create or determine the whole world.
Again you're arguments seem to conflate the concept of our knowledge of reality with the concept of reality.
Yes, our knowledge, and therefore whatever model of reality is based on that knowledge, can only ever be finite. There might be unknowable aspects of reality, but given that they are unknowable, speculation on them is moot.
Quoting Cabbage Farmer
My point was exactly that anything that is empirically knowable must be finite. I further contend that "the universe" should refer to something empirical, as a matter of practicality.
Quoting Cabbage Farmer
I just don't see why whether the "world behind the world" is or is not finite is "relevant".
Quoting Cabbage Farmer
I do not agree with that. I am a constructivist, so yes I do claim that, in a way, our knowledge creates reality. Not necessarily "in every regard" though, since I am not sure what you wish to imply with that.
Quoting Cabbage Farmer
They may not create the world, but they nevertheless populate it with all the content. All we can say about the world absent experience is that it exists.
Wasn't this enough?
Oh right, it's a Philosophy forum. :wink:
This property has been conjectured for pi and certain other constants, but it has not been proven. In any case, knowing that a certain sequence is buried somewhere in that infinite stream is not as helpful as it might seem, because on average, the index that points to the beginning of the sequence that you are looking for would be so large that it would contain more information than the sequence itself. Think Borges's The Library of Babel. Anyway, this is indeed fun to think about, and the above mentioned conjecture has kept number theorists busy.
It is not a conjecture. There is proof that between two real numbers there is always another real number. That is valid in base 10 and, in Spanish, in base 27 (just assign a number for each letter). In English, in base 26. But it is not a conjecture. About the theorem cited, see the following:
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
However, the hypothesized property of pi to which you were referring - that it contains every finite sequence of digits - does not follow from this elementary property of real numbers. This would actually be a weaker version of absolute normality - the property of containing every finite sequence of digits in every base with "equal frequency" (scare quotes because this is more complicated than it sounds). While it is has been shown that "almost all" numbers are absolutely normal, it is surprisingly difficult to prove this property about a specific number. As far as I know, this has not been proven about any known number, including pi, although experimentally it has been confirmed for its calculated digits.
https://en.wikipedia.org/wiki/Transcendental_number
Interesting. Thanks.
I believe I've read that Chaitin's Omega is known to be normal. It's not actually a specific number, just a class of numbers; and you can't write down any particular one, since none of them are computable. So you may or may not take this as a counterexample to the statement that no particular real is known to be normal.
https://en.wikipedia.org/wiki/Chaitin%27s_constant
https://cs.stackexchange.com/questions/67695/chaitins-constant-is-normal
Not at all. You could still be right. At best the Chaitin number shows that there's a normal number among a class of numbers we can define in first-order logic. But we already knew that normal numbers exist. So it would still be fair to say that we don't know a particular one.
Specific speculative claims about what is unknowable are unwarranted conjectures.
It seems to me that claims to know the actual limits of the world -- as distinct from the limits of the "known world" -- are instances of such speculative claims. In my view you have signed on to such a claim, seemingly without realizing that this is what you are doing. I agree such claims are unwarranted, which is why I've been objecting to them here.
By contrast, the claim that it seems we cannot know whether some facts or features of the world are unknowable in principle for creatures like us is arguably not speculative at all. It's not an empirical or metaphysical claim about what the world is like. It's an epistemological claim that seems to follow from any reasonable conception we might assign to terms like "know" and "world".
Quoting Echarmion
I'm not sure I would agree with this.
The fact that we "know something" or "know about something" -- a dog or a table, for instance -- does not entail that we know everything about it. It's not clear that we ever have "complete knowledge" of a thing we know or know about, or what it might mean to say that we do have complete knowledge of a thing.
Accordingly, I see no reason to object to the claim that partial knowledge of an infinite thing would count as knowledge. So, if the world is infinite in some respect, say in space or time, and our knowledge of it is finite, this would not entail that we don't know the world, but only that our knowledge of the world is partial and incomplete. But in this respect it would resemble our knowledge of many "finite things", like dogs and tables.
Quoting Echarmion
The claim we began by addressing is a claim to have proved that "there is no infinity". I take it you and I are still considering that claim when we use words like "universe" and "world" in this conversation.
What is the practical value of a conversation about whether "there is infinity"?
On my view, at least part of the practical value is that it directs our thoughts to consider the limits of our knowledge of the world.
Accordingly, I reject your ad hoc definition on practical as well as theoretical grounds.
Quoting Echarmion
I do not claim there is a "world behind the world". I say, by definition, there is one world; and it seems that world is knowable at least in part, on the basis of appearances.
My reply to your remark about practicality should suffice to indicate my position on the matter of relevance.
I'll add this: If it's true, then it's relevant. It's my aim to practice clear, coherent, and honest speech in philosophical conversation, and to offset our tendency to error, confusion, and insincerity.
It seems to me that philosophical confusion, even in small and abstract matters, may have far-reaching personal and political implications.
I would characterize philosophical discourse as pursuit of a sort of personal and political harmony.
Quoting Echarmion
In what way do you say our knowledge creates reality?
I agree we have a peculiar way of participating in the world as sentient animals and as cultural animals with powerful conceptual capacities. I suppose we can say each sentient animal "creates" its way of participating in the world just by living, and this participation includes a way of perceiving the world and a way of acting in the world.
I see no reason to say that to perceive a world is to "create a world", nor that to perceive a dog is to create a dog. Nor that to understand a state of affairs is to create that state of affairs, nor that to run into a wall is to create that wall. And so on. So far as I can tell, that would be getting carried away with talk of our "creativity".
Quoting Echarmion
Do you mean to say that experience and scientific method "populate the world with all the content" of the world? What does it mean to say this?
Does it mean that when we perceive a dog, our minds somehow "populate the world" with a dog that in fact does not otherwise exist in the world, or with a dog that in fact is not otherwise "contained" in the world?
Here again it seems you may be conflating a conception of the world as it is in fact with a conception of our knowledge of the world.
Do you suppose the dog is not "in the world" and "does not exist", unless we know it?
I think I agree with this. I just don't think claiming that infinity cannot exist in the "world" is a specific claim in that sense. It's a general statement about the concept of infinity.
Quoting Cabbage Farmer
But even if "the world" is infinite in some respect, it is impossible to ever observe it in that infinity, since observing the infinite is, by definition, never complete. Even if we observed a seemingly endless succession, all we can ever say based on that is that the succession is indefinite.
So if we extend the meaning of "the world" to things that are, in principle, impossible to observe, we are doing unwarranted metaphysical speculation.
Quoting Cabbage Farmer
The problem is that the claim "there is no infinity" isn't specific enough. Normally, when we talk about what is and isn't we're talking about physical reality, not theoretically possible worlds. We can discuss possible worlds, but I am not sure the conclusions would be very interesting.
Quoting Cabbage Farmer
Right. But you do see that my argument is essentially an epistemological one, so I does fit your definition of practical value.
Quoting Cabbage Farmer
So do you agree infinity belongs to the part of the world that's not knowable?
Quoting Cabbage Farmer
An interesting perspective. Of course, what "true" even means is itself a matter of philosophical discourse.
Quoting Cabbage Farmer
Well I think that there are neither dogs nor walls outside our heads. Those are categories put together by human minds on the basis of certain patterns in the information that these minds receive. There is some source of that information, but it's the mind that constructs the patterns. The patterns are not the information though, like the pattern on a sweater is not the fabric.
Quoting Cabbage Farmer
If we abstract from all experience, talk of "the world" is meaningless. Fundamentally, the world is the collection of our knowledge about the world. We can reason, based on experience, that there are things we don't know about the world, and from that we get the idea of a larger world of which we know only some part. So the "conception of the world" is actually secondary and based on experience.
So if you lived on some undiscovered island and had never seen or heard of a dog, and neither did anyone you had ever seen or heard of, you might still know, based on experience, that you sometimes discover new animals. And hence you might say that "unknown animals exist". But would you say that "dogs exist"? You wouldn't, because the term "dog" is meaningless to you. Someone would have to explain a dog based on things you have already experienced for the claim to make any sense.