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.
I'd argue that nowadays there are way fewer the mathematicians who are supporters of mathematical platonism (I include here also Popper's view that mathematical sentences are our own creation but they exist independent of us in a world of culture, his 'Third World')...Formalism about numbers is far more popular at least from Hilbert onward. Sure Hilbert's program does not work till the end (Godel's legacy) but the formalist programe is still tenable to some extent. In this view syntactically correct formulas which are undecidable in a certain set of symbols/axioms/rules of inference are not true/false by default (some say 'true/false but indemonstrable' about undecidable statements). It is us which make them true/false either by adding such statements (or their contraries) as new axioms or by modifying the initial set of axioms so that these undecidable initially formulations appear as theorems in the new system (that is they are deducible from the new set of axioms).
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.
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.
The formalist has no problem dealing with infinite sets. After all, "infinite" is just an adjective with a formal definition. Let me clarify: in any model of the integers, there exists true statements that cannot be proved from the axiom of the integers. 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.
When I talk about "the integers", I mean the "standard model", 1, 2, 3 ... My point is that in the end there is no way to formally define this, you have to have some Platonist notion of the (standard model of the) integers being "out there"
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).
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.
That's wrong. In Zermelo set theory, it goes like this: (Recall that in the usual model set-theoretic 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: Code (Text): 1 --> N --> N | | | | | | V V V 1 --> A --> A with both arrows N --> A the same morphism. (This diagram encodes the idea of recursively defining a function N --> A)
No, I don't buy that. 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.
There are a couple of questions here, however, that do depend on philosophical beliefs. 2+2=4 may be true for everyone, but is "2" the number a real thing or "just a concept"? Is that even a true dilemma, or can something that is a concept be real? For some people, a number is only real if there is something in physical reality that occurs in that quantity. There being 2 apples is, for some people, proof that 2 is real. So what exactly are number (and other mathematical objects), are they real, and in what sense? The other question is that of mathematical propositions. Are they purely man-made, 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?
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 first-order 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 first-order ZF. By the way, when cast in the language of sets, the completeness axiom is a first-order 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 first-order 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 first-order 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 first-order 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.
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.
So true! Mathematical talent sometimes seems more like what the Ancient Greeks thought of Eros: a cruel and irresponsible god who takes you over willy-nilly and gives you deep personal rewards while ruining your social life.
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?
I think the formalist dream (Hilbert's programme) has turned out not to work. Sure, For All Practical Purposes (FAPP), it is still a useful tool, but it has its limits (Goedel). 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 ?
Well, don't take this too seriously - I haven't given it much thought. Take, say, the (finite number of) symbols of English (alphabet, punctuation symbols, etc.). Imagine all possible strings of these symbols, say, 36 symbols long. Some of the strings will be grammatical English sentences (or words, phrases, etc.), even though I didn't include any rules of English grammar in the system. This will be one of those sentences. Is that not a pure coincidence? If not, what would make it a pure coincidence? 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?
do you have any evidence the "most well known mathematicians" are platonists? Can you even name me any non-dead ones? Of course whatever you views on what maths is in this sense don't affect your ability to do it, and I would hazard a guess that in a world post Goedel, in which the continuum hypothesis and the axiom of choice are independent of the axioms of ZF, in which it is possible to construct a theoretical game in which both players always win, where the Banach Tarski paradox holds, then we are more likely to be formalists of some description, I know I am (not that i am at all a "good pure mathematician"). Indeed I only know of one other mathematician's declared view, and he (Gowers, Field's Medal) is certainly not a platonist. http://www.dpmms.cam.ac.uk/~wtg10/philosophy.html
Roger Penrose is a declared Platonist. And Goedel was one too, according to Rebecca Goldstein's new book about him, Incompleteness. She claims his famous theorems were an attempt to defend the Platonist view from Hilbert's "abstract game with marks on paper" attack. A successful attempt which was improperly co-opted by the "man is the measure of all things" crowd.
Apart from Penrose do we have any living declared PLatonists out there? (Hawking is allegedly a positivist if that is a counter balance to Penrose.) Bear in mind that the nature of mathematics has changed dramatcially over the last 100 years. In fact let me slightly rephrase it as: apart from Penrose who is 'famously' a platonist, and Gowers who just happens to have written a talk about philosophy and mathematics where he reveals he isn't a platonist, has any modern 'big' mathematician actually ever said what they think? I guess 'no' and this underlines the fact that whatever your view of it, if you even have a view, doesn't matter one jot. Although I would put myself as a formalist of some kind I can't say I have ever thought about it in anyway other than to decide, after some prompting like this, that I do not consider there to be any reason for platonism to be 'true', indeed you might make a case that it is less that I am a formalist than simply "not a platonist".