
#1
Jul505, 07:47 PM

P: 2,268

Can anyone name any good or famous pure mathemaicians who does not believe in mathematics existing in a world indepedent of humans? In other words, are there any pure mathematicians who believe that the rules of mathematics are purely made up by humans, although there is many mathematical properties to discover winthin the (invented) rules or axioms.




#2
Jul605, 03:37 AM

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#3
Jul605, 04:10 AM

P: 499

I would say that a belief in actually infinite sets requires a Platonist point of view, and since Cantor many people have had this belief. Hilbert, despite his formalist views, talked of "Cantor's Paradise". If you're a real formalist then you think of dealing with infinite sets as just a manipulation of various symbols.
Godel told us that there are statements about the integers which cannot be proved, but which we nevertheless know to be true. Indeed without admitting to such knowledge its difficult to say what we mean by the integers. (see http://www.chronon.org/articles/the_integers.html) Of course the situation is never simple. Penrose is well known as a Platonist, but he seems to steer away from the questions of infinite sets, prefering to base his ideas on a more geometric viewpoint. 



#4
Jul605, 06:09 AM

P: 261

Good Pure Mathematicians who are not Platonists?
The formalist could use Russell's definition of a number (classes of classes) and argue that all we need to define let's say natural numbers is an intensional description though the extension is not listable (extension can always be reduced to intension, the reverse is not always valid). In this view natural numbers are not seen as Platonic, the empirical correspondence with phenomenal facts (observed facts) of some undecidable statements being contingent.




#5
Jul605, 06:47 PM

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The formalist has no problem dealing with infinite sets. After all, "infinite" is just an adjective with a formal definition.
It's not as impressive as it sounds (though it's still an impressive result): Gödel's theorem proves that there exist statements in the theory of the integers that cannot be proven or disproven from the axioms of the integers. Since one of the things a model does is that it gives a way to assign to each statement a truth value, we must have a true statement that cannot be proven from the axioms of the integers. 



#6
Jul705, 12:50 AM

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#7
Jul705, 03:31 AM

P: 261

I think we must make a clear distinction between the truth (of mathematical statements) as formalists understand it (true mathematical statements are axioms or can be deduced from them) and truth understood as conformity with some empirical facts.
Formalists hold that mathematical statements have no meanings, we must attach a specific semantics when use mathematics in physical sciences. So that a mathematical statement can be undecidable in a certain system of axioms of mathematics and still to be 'true' in relation with empirical facts as much as we attach the required semantics. Schrodinger's equation is a good example here (it was 'invented' by Schordinger using an analogy with classical physics). If we add Born's intepretation of the wavefunction we can say that it is 'true' in relation with empirical facts (it is part of the system of axioms used by science, held, at least provisionally, as true) though it is undecidable in the system of axioms of mathematics (at least it was so in the 1920s). I think the problem of numbers falls in the same case, there is no necessary link between the two types of truths. All we need to talk of mathematical truth is an intensional definition of numbers within formalism, otherwise we should endorse Kant's view that there is apriori synthetic knowledge (not the best of options in my view). 



#8
Jul705, 03:59 AM

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P: 2,330

I'm rather confused by this thread. The math itself is the same for everyone, regardless of their philosophical beliefs.
As for the OP, you can search for formalists and logicists. They include Hilbert, Carnap, Tarski, Curry, Frege, and many others. I'm not sure about intuitionist and constructivists  some may agree with platonism in ways. 



#9
Jul705, 06:07 AM

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(Recall that in the usual model settheoretic of the natural numbers, 0 is modelled by the empty set, and the successor operation is modelled by x → x U {x}) Axiom of Infinity: There exists a set S such that [itex]\emptyset \in S[/itex] and [itex]x \in S \implies x \cup \{ x \} \in S[/itex]. Then we define N to be the smallest set having that property. (It takes a slightly roundabout method to actually say "the smallest set having that property", though) In group theory, one could say: Z is a free commutative group with one generator. In ring theory, one could say: Z is the initial object for the category of rings: for any ring R, there is a unique homomorphism Z → R. In topos theory, it goes like this: A natural number object is an object N and arrows 1>N and N>N (called "zero" and "successor") such that, for any other pair of arrows 1 > A > A, we have a unique commutative diagram:
(This diagram encodes the idea of recursively defining a function N > A) 



#10
Jul705, 09:50 AM

P: 499

If you just use first order axioms then you won't be able to uniquely characterize the "standard model" of the integers (Godel tells us that there will be nonstandard models of your axioms) And higher order axioms will correspond to a Platonist notion of actually inifinite sets or the like. For instance in the real number axioms the real numbers (and the integers within them) are defined uniquely, but the completeness axiom is 2nd order, requiring arbitrary infinite sequences of rationals. 



#11
Jul705, 10:02 AM

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The other question is that of mathematical propositions. Are they purely manmade, and hence is their truth contingent, or are they necessary truths? In other words, is the "world" of mathematical propositions created and contingent on the mind of man, or is it simply accessed by our minds while it exists independently? 



#12
Jul705, 06:25 PM

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I don't see why higher order logic corresponds to Platonist notions  to a formalist, it's still just a bunch of symbols.
I don't see why the possibility of nonstandard models matters either, to any type of mathematician, but especially a formalist. Within firstorder ZF, you still have the theorem that any two models of Peano's axioms are isomorphic... it doesn't matter that there may be different models of firstorder ZF. By the way, when cast in the language of sets, the completeness axiom is a firstorder statement: [tex] \forall S, T \subseteq \mathbb{R}: (\forall s \in S, t \in T: s \leq t) \implies \exists x \in \mathbb{R}: (\forall s \in S, t \in T: s \leq x \leq t) [/tex] The corresponding formulation in topos theory would also be a firstorder statement. On the other hand, if you're working in the language of ordered fields, it is a second order statement (because S and T range over firstorder unary relations on R, rather than subsets of R), but it still does not uniquely pick out a model. The completeness axiom is satisfied by all real closed fields (such as the real algebraic numbers) The proof is by Tarski's theorem: For any particular S and T, the specialization of the completeness axiom is a firstorder statement, and is true in the reals, therefore it's true in every real closed field. Since S and T were arbitrary, the conclusion follows. 



#13
Jul905, 11:21 PM

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philosophical beliefs probably have nothing to do with mathematical ability or any other kind of ability. remember the movie about mozart and salieri? the irony was that the crude but brilliant mozart was a better composer than the devout nutcase salieri.




#14
Jul1005, 11:50 AM

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#15
Jul2105, 02:06 AM

P: 2,268

I think that if a Mathematician was a Platonist than he/she would have more of a motivation to study mathematics thereby enjoy it more. As a result he might also spend more time on the subject and so has a higher liklihood of being more superior a mathematician than his non Platonist collegue. This may explain why most well known pure mathematicians are Platonists. Even Hilbert showed signs of it when he praised Cantor's work so highly. How can one be both a formalist and Platonist at the same time?




#16
Aug205, 07:13 AM

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And in fact that shouldn't surprise us. After all, formal logic is nothing else but a game of finite sequences of a finite set of symbols that satisfy certain formal rules. Formal logic is the study of a certain set of mappings which map {1,2,...n} into, say, the ASCII set. It only has a meaning when we GIVE it a meaning  and that's Platonism, no ? I'd even say, for this formal game to have a meaning in the first place, you'd already have to ASSUME certain properties of natural numbers, in order to be able to say things about these mappings from {1...n} into the ASCII set. So using then this formal game to PROVE properties of natural numbers seems to be a bit circular IMHO, because you've USED certain properties of the natural numbers to set up the formal machinery in the first place. For instance, how do you prove that every natural number has a successor, purely formally ? 



#17
Aug205, 12:24 PM

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Why can't you think of a theory as just having been chosen from all the possible options?  And in what way, if any, would the choice not be arbitrary? 



#18
Aug205, 12:53 PM

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http://www.dpmms.cam.ac.uk/~wtg10/philosophy.html 


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