Hi. I’m Adam, from central Florida, a graduate of Brandeis Uni, now a tutor for Kaplan until I get into grad school. I study philosophy :dough: , history 
, and increasingly more mathematics 
… which means I am the model from which all nerds are forged.
All the best,
Adam
Welcome to the forum, “model”. - softwareNerd
How much math have you studied so far, Adam? Are you learning about the math itself or about philosophical questions about mathematics?
I haven’t taken the normal progression of math classes because every time I try to sit through a normal lower-level math class I have to fight to control projectile vomiting. I hate having to memorize formulas like catechisms, so I never took much math in college. I’ve taught myself up to trig by actually learning concepts, and I’m just now finishing that and broaching calc. But I have approached the rest of math through the backdoor via symbolic logic, mathematical logic, and philosophy of math. Eventually I hope to teach myself through to set theory, proofs, complex numbers, and a little application in stats, probability, and some (or a lot, depending on how I like it) physics.
Good idea to learn it yourself.
But before you really can understand mathematics (especially the concept of “infinity” as simply a concept of method) you will first need to finish your introduction to Calculus. Hint, have you studied the nature of limits yet?
Not knowing the higher levels of mathematics hasn’t prevented me from understanding Frege’s The Foundations of Arithmetic, Benacerraf, and Carnap (and your statement about method is, granting that you make no distinction between “method” and “system”, liable to the same criticism that Benacerraf is liable to). I see what you mean by “infinity isn’t an object, it’s a method” I just don’t see how it will, if indeed it can, avoid Platonism.
And I have read a rigorous definition of limits, which I understand, though I haven’t yet worked with them.
So then you understand in simple mathematical language that when you say take the following limit-- 0/x as x goes to infinity that its LIMIT is infinity but that this does NOT imply the actual existence of a point known as “infinity” since this would imply a contradiction. Not only that but when used this way “infinity” implies a stolen concept because as I said earlier numbers increase without bounds without implying an end (infinity) (which would also be a contradiction in terms) since such a point would imply that the numbers are bounded. Not only that but an “infininte point” itself is a contradiction and a metaphysical monstrosity.
Right. Now on to Platonism.
Again, no offense, but perhaps you do not understand the response I made to your statement that infinity and mathematics are “methods”. It is common in mathematics for people to brush philosophy aside saying math is all just a “system” and think that they’ve left the matter in the dust. Philosophers are not nearly so easily sated, though. The problem arises, what is this system? In the same way that philosophers ask, “Is a number a genuine object?” we ask, “Is a system a genuine entity?” Benacerraf tried to make the case that mathematics is a system where numbers are not objects–that they are nothing. Numbers can be anything. They can be spoken words, “One, two, three…” or written marks, "1, 2, 3… (or 0, {0}, {{0}}… or 0, {0}, {0, {0}}…), or even sticks and stones. The naturals are just those things that we use to line up representations (e.g. words or symbols) to the represented (e.g. five cows) in a one-to-one correspondence. Benacerraf, all the same, is liable to the question, “And this general practice… What is it? Are these rules hovering over us, will we be struck with a heart-attack if we violate them? How does the system account for infinity? Does there exist only so many numbers as you can produce representations? If you say that infinity is the number of things you could count, were you to have an unlimited supply of representations, then you have assumed infinity into your argument and the whole point is to try to find a way out of calling it an axiom.”
Carnap tried to push the line further back. He said this talk of a system misses the point. Math is just a game we play, and we can play any game we want. We just happen to prefer the math game in some contexts. If you want to play another, go ahead. People do it all the time when they speak English, which is also just another game. Math is not a system, really–rather, you should think of it as a linguistic, social practice. The nice part about this Carnapian approach is that it has a very consistent answer to infinity: Infinity is just part of the math game. It’s like home-runs in baseball. Home-runs don’t exist outside of baseball–that would just be a ball flying over a fence. But because of the rules of baseball, there are home-runs in the game. The rules of the game of math simply prescribe that there is no greatest number. In counting, “1, 2, 3…” there is only the language practice of writing “…” That’s all that infinity is. Yet again, however, you can ask Carnap, “So what are these games? Are they genuine entities? Why is it that this particular game is so successful in living life and firing rockets? Do the individual games live within the greater category of games, and is this category a genuine object? What happens when we stop playing this game? Does math disappear?”
As you can see, almost every move you might throw your weight behind seems to be liable to the same Platonic critique. In my personal philosophy, I have managed to avoid and expunge all Platonism except, perhaps, for this one vestige.
Mathematics is a direct result of observing the real world and the logic of how it interacts… this is not the result of some Platonic forms existing (where?) but of the objective nature of reality. I don’t think this a problem of me not understanding your question but of you not understanding the responce. My main answer would be to read more on Objectivism and you will understand how many of your questions are just a result of rationalism and that Platonism can NOT be correct because it sneaks in the supernatural. I.e., things that can NOT be shown to exist in reality and/or are arbitrary in nature.
I re-read my last post and realized how tired I must have been when I wrote it because it is unclear and my limit was wrong. It should have been 1/x or n/x with n being any natural number and not 0/x which would become undefined as x –> 0.
I’ve read quite a bit of objectivism, and I still don’t see how you’ve discounted Platonism. First, while he has a very supernatural feel, it is quite easy to interpret Plato has having a rationalist spin. Have you read Plato? Moreover, if you were to discount Plato you would not have provided an account of numbers. If anything, were you to disprove Plato, since all roads thus travelled seem to lead to Plato, it would sooner appear that you’ve proved a contradiction.
So as the tally stands, Platonism has not been proved nor has it been disproved. All thus-far conceived contraries of Platonism have proved contradictory, but there is no evidence that this is a complete list of non-Platonist perspectives. So it should be clear that any answer to the question has to be original or Platonic. Which it is, I do not know.
Also, 1/x wouldn’t work either if you permit x to be 0, since that would make it undefined. I’m not sure what you’re saying by, “not 0/x which would become undefined as x –> 0,” since 0/1 is defined. It’s 1/0 that is not defined, unless what you meant was that 0/x could turn out to be 0/0, which is also undefined. Perhaps you have some special use of ‘–>’, because for me it is a logical symbol meaning ‘if… then’.
Welcome, Aleph. (Is Hebrew one of your languages?) I have a few comments that might interest you.
I’ve read quite a bit of objectivism, and […]
In traditional history of philosophy, the term “objectivism,” as I have seen it defined, names a particular thesis: a belief that there is a reality independent of consciousness. (See, for example, the definition in the Routledge Encyclopedia of Philosophy, at least in the edition that was current a few years ago, which was the last time I examined it.) That is, of course, a metaphysical tenet, not a whole philosophy.
On the other hand, “Objectivism,” with a capital O, is the proper name for a whole philosophy, the philosophy that Ayn Rand created. See “Objectivism,” The Ayn Rand Lexicon. You might also examine the Forum Rules, for capitalization.
[…]it is quite easy to interpret Plato has having a rationalist spin. […]
I am not sure what you mean by “spin” here, but you are definitely right that Plato’s philosophy is rationalist, if that is what you are saying. Objectivism, of course, rejects rationalism and other corrupt epistemologies – as well as everything built on them.
In the context of Objectivism, which is the context for this forum, “rationalism” is the term that names an epistemology that emphasizes syllogistic rigor – but starting from arbitrary premises. As Ayn Rand describes it, rationalism is the claim “that man obtains his knowledge of the world by deducing it exclusively from concepts, which come from inside his head and are not derived from the perception of physical facts …” (See “Rationalism vs. Empiricism,” ARL.)
P. S. – Aleph, I have been wondering what you mean by your subtitle:
“A Pleasure To Make Your Acquaintances … Except You.”
Would you explain? I often miss visual clues to such puzzles. Have I done that again?
I re-read my last post and realized how tired I must have been when I wrote it because it is unclear and my limit was wrong. It should have been 1/x or n/x with n being any natural number and not 0/x which would become undefined as x –> 0.
0/x tends to 0 as x->0 (think about it in terms of epsilon-delta). The whole point of limits is that the function doesnt need to be defined at the point youre evaluating the limit (f(x) = 0/x is discontinuous at 0 where it has a removable singularity, but the limit at both sides is 0).
But anyway, the way that infinity is treated within standard analysis is different from the way its treated within set theory (this is the distinction between potential and completed infinities, and why Cantor’s work was so controversial). In the context of analysis, we would say something like “The fact that there are infinite integers means that the sequence x_n, where x_n takes integer values and x_n+1 > x_n for all n, has no limit”. And this captures what we mean by unboundedness. But when doing set theory, we talk about the completed set of natural numbers, N. And this is not meant as a potential infinity. The Axiom of Infinity used in ZFC states, quite unambiguously, “There exists at least one infinite set”.
On a sidenote, you dont need the limit concept to do calculus - you can do it using infinities and infinitesimals like Newton and Leibniz did. Calculus was originally founded upon infinite(simal) numbers, and the reason why the limit method is taught in undergraduate analysis classes today is because of historical coincidence (a rigorous formulation of calculus using limits was found around a century before a rigorous formulation using infinitesimals), not one of logical necessity. Limits dont show us that infinity doesnt exist mathematically, they show us that we dont need to postulate infinity in order to do calculus (ie we can choose to include it in our system, or not). Of course, none of this means that ‘infinity’ exists in the real world, if this statement is taken to mean that you can ‘infinitely many objects’ (although you cant have -4 objects or 3+2i objects either).
this does NOT imply the actual existence of a point known as “infinity” since this would imply a contradiction
Actually, defining a point known as infinity is useful, and often done. You need it to construct the Riemann sphere for example, and its fundamental to projective geometry.
Right. Now on to Platonism.
Carnap tried to push the line further back. He said this talk of a system misses the point. Math is just a game we play, and we can play any game we want. We just happen to prefer the math game in some contexts. If you want to play another, go ahead. People do it all the time when they speak English, which is also just another game. Math is not a system, really–rather, you should think of it as a linguistic, social practice. The nice part about this Carnapian approach is that it has a very consistent answer to infinity: Infinity is just part of the math game. It’s like home-runs in baseball. Home-runs don’t exist outside of baseball–that would just be a ball flying over a fence. But because of the rules of baseball, there are home-runs in the game. The rules of the game of math simply prescribe that there is no greatest number. In counting, “1, 2, 3…” there is only the language practice of writing “…” That’s all that infinity is. Yet again, however, you can ask Carnap, “So what are these games? Are they genuine entities? Why is it that this particular game is so successful in living life and firing rockets? Do the individual games live within the greater category of games, and is this category a genuine object? What happens when we stop playing this game? Does math disappear?”
This is not what Carnap said, in fact its in direct contradiction to his formalist view of mathematics as just being a formal system. I think youre confusing him with (eg) the later Wittgenstein.
Carnap would have agreed that infinite only comes into mathematics when/if we introduce it into our symbolism, but I think he would have disagreed with most of the other claims youve attributed to him (most notably “this talk of a system misses the point”, and “Math is not a system, really–rather, you should think of it as a linguistic, social practice”).
So as the tally stands, Platonism has not been proved nor has it been disproved. All thus-far conceived contraries of Platonism have proved contradictory, but there is no evidence that this is a complete list of non-Platonist perspectives. So it should be clear that any answer to the question has to be original or Platonic. Which it is, I do not know.
What do you mean by ‘proving Platonism’? What sort of thing would count as evidence for it?
Anyway, its incorrect to say that the contraries of Platoism have proved contradictory. For instance, Formalism is perfectly consistent, its just pointless and ducks the interesting questions. Intuitionism is consistent, its just very complex and cant reconstruct all of classical mathematics. None of the more empirical positions which have become more popular recently (eg those espoused by Polya, Putnam, etc) are contradictory either (and imo they are a lot closer to the truth than any of the traditional approaches - I would recommend this as an excellent introduction to ‘quasi-empiricist’ philosophies. On a sidenote, while Objectivism doesnt have a philosophy of mathematics as such, I would tentatively suggest that this has more in common with what I see as being the ‘spirit of Objectivism’ than the traditional approaches do)
Also, 1/x wouldn’t work either if you permit x to be 0, since that would make it undefined. I’m not sure what you’re saying by, “not 0/x which would become undefined as x –> 0,” since 0/1 is defined. It’s 1/0 that is not defined, unless what you meant was that 0/x could turn out to be 0/0, which is also undefined. Perhaps you have some special use of ‘–>’, because for me it is a logical symbol meaning ‘if… then’.
-> means ‘tends to’, its used to denote the limit of a function/sequence. Its the value which the sequence is getting closer and closer to, as the variable obeys some condition. For instance, the function f(x) = 1/x tends to 0 as x tends to infinite, because as x gets bigger and bigger, f(x) gets smaller and smaller. Hence we write “f(x) -> 0 as x-> infinity”. Similarly, we would write “f(x) -> infinity as x-> 0”, since 1/x becomes larger and larger as x gets closer to 0.
aleph_0:
You mentioned before that you have read some Objectivist literature. Have you read Introduction to Objectivist Epistemology? If you’re looking for some of Ayn Rand’s thoughts about mathematics, that would be the place to look.
Hello BurgessLau, yes, I speak a bit of Hebrew.
On the other hand, “Objectivism,” with a capital O, is the proper name for a whole philosophy, the philosophy that Ayn Rand created. See “Objectivism,” The Ayn Rand Lexicon. You might also examine the Forum Rules, for capitalization.
Fair enough, I’m in the habit of not capitalizing beliefs, but very well then. I believe everything I have posted up to now that reads “objectivism” should then read “Objectivism”.
I am not sure what you mean by “spin” here, but you are definitely right that Plato’s philosophy is rationalist, if that is what you are saying. Objectivism, of course, rejects rationalism and other corrupt epistemologies – as well as everything built on them.
I’m aware, I’m just pointing out that Platonism need not be seen as religious.
P. S. – Aleph, I have been wondering what you mean by your subtitle:
“A Pleasure To Make Your Acquaintances … Except You.”
Would you explain? I often miss visual clues to such puzzles. Have I done that again?
I’m just playing.
Carnap would have agreed that infinite only comes into mathematics when/if we introduce it into our symbolism, but I think he would have disagreed with most of the other claims youve attributed to him (most notably “this talk of a system misses the point”, and “Math is not a system, really–rather, you should think of it as a linguistic, social practice”).
I cannot find my old Carnap book, so we’ll leave it unattributed to him for now, although he was writing specifically in criticism of the Benacerraf approach of systems, precisely because the system still seemed to be a Platonic form. That is not to say that he thought math could not be called a system, but merely that such an explanation is not complete. To brush it off as a “system” misses the point of denying Platonism.
What do you mean by ‘proving Platonism’? What sort of thing would count as evidence for it?
First, I haven’t claimed to have proved it. What I “mean” by proving it is the same as I mean by proving any philosophical point. Providing the most simple explanation of observed and logical phenomena that does not contradict itself or the phenomena. What would constitute a proof of it I’m not perfectly sure–if I were I wouldn’t need to ask about it. Still, a proof could possibly look like, “Either there is an objective entity that is the system of math or there is not. If we assume there is not, we necessarily reach a contradiction, thus, the system is a genuine object.” Other forms of proof might also apply.
Anyway, its incorrect to say that the contraries of Platoism have proved contradictory. For instance, Formalism is perfectly consistent, its just pointless and ducks the interesting questions.
Formalism, as I understand it, still treats the system of math–whether they recognize it or not–like an object, thus denying the premise of non-Platonism.
Intuitionism is consistent, its just very complex and cant reconstruct all of classical mathematics.
I’m not sure I know what intuitionism is. Perhaps it’s the answer.
None of the more empirical positions which have become more popular recently (eg those espoused by Polya, Putnam, etc) are contradictory either (and imo they are a lot closer to the truth than any of the traditional approaches - I would recommend this as an excellent introduction to ‘quasi-empiricist’ philosophies.
Isn’t Putnam’s account something like Mill’s? If not, I’ll check it out. If it’s like Mill’s, I would argue it too is inconsistent–perhaps not internally, but clearly inconsistent with the practice of mathematics.
This thread turned into a debute, heh? :lol:
Although I don’t consider myself a member, I’m pleased to see more people with ideas come.
You mentioned of a “geek” concept. Too bad it is negative one in USA, when it is obvious to both of us that a “geek” is to be the positive term.
Offtopic, I came from a Post-Soviet country (Ukraine), where concept such a “geek” doesn’t exist as one word in the slavic languages. As I recall myself in schools there, getting awesome grades is a way to be popular. (Compared to what you know and might view yourself as, I know you’ve head, but I don’t think you’ve been pointed at the opposite state of affairs in other place of the world) Why does this make me angry? Oh, yeah, reading Atlas Shrugged.
P.S. Please do not answer to me here for another debate. I will not answer here, in order to keep this purely as your introduction.