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Is mathematical practice:

  • an act of discovery of eternal objects and ideas independent of human existence;
  • an intuition-free game in which symbols are manipulated according to a fixed sets of rules;
  • or a product of constructions from primitive intuitive objects, most notably the integers?

I would like someone to explain what schools of thought are behind these definitions, what is relation between them, can all be equally valid, is there the most accurate definition among them, and all related questions...

I am just a laymen interested in philosophy.

VividD
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    For teh frist one, see [Platonism](http://plato.stanford.edu/entries/platonism-mathematics/); for the second see [Formaism](http://plato.stanford.edu/entries/formalism-mathematics/) and for the third onesee [Intuitionism](http://plato.stanford.edu/entries/intuitionism/). In general, see [Philosophy of Mathematics](http://plato.stanford.edu/entries/philosophy-mathematics/). – Mauro ALLEGRANZA Oct 12 '14 at 13:38
  • And there are also moder recent issues : see [Naturalism](http://plato.stanford.edu/entries/naturalism-mathematics/) and [Indispensability Arguments](http://plato.stanford.edu/entries/mathphil-indis/). – Mauro ALLEGRANZA Oct 12 '14 at 14:55
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    Even formal systems admit intuition: that is the difference between a novice and an expert at chess, for example. One must merely be honest about where the rules are coming from, and what we hope to accomplish by 'playing'. – Niel de Beaudrap Oct 12 '14 at 14:56
  • It's the stuff between philosophy and physics. – user4894 Oct 12 '14 at 18:21
  • Your question is far too general and demanding in detail to allow for a reasonable answer to be given here in under 400 pages. Try to choose a more specific question, and maybe try posting multiple questions. Focus on one school of thought or ask how a specific issue relates to each different school of thought. – nwr Oct 13 '14 at 01:14
  • @NieldeBeaudrap Yes, but formalists see intuition as primarily the ability to unconsciously compute what one could either consciously compute or reasonably guess. It does not mean the same thing to the rest of us. For Platonists and Intuitionist, confirming, grounding or elaborating intuition is the point of the exercise of mathematics. It is what gives the system meaning. –  Oct 13 '14 at 18:21
  • Mathematics is a form of thinking. No need to idealize it. No man thinks of mathematics before death. Every man thinks before death about love, god, revelation and sufferings. Think about **THAT**. Mathematics and science is a form of recreational learning of **how** to think. – Asphir Dom Oct 13 '14 at 20:03
  • Just for the record all objects are eternal. So sweeping in the yard while noticing and thinking is pretty divine too. Did you try? – Asphir Dom Oct 13 '14 at 20:06

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That's the golden question! And, by the course of things, without solution. The answer pressuposes some philosophical background which is practically based on opinion. A good approach to the schools are http://plato.stanford.edu/entries/philosophy-mathematics/. I would also recommend the preface to the second edition of https://archive.org/details/principlesofmath005807mbp. In choosing a school of thought, don't forget to consider that every theory by it's essence is fallacious; for example, the theory of concatenation has logical circularities by it's own nature, because we use concatenation to approach the theory (a word in english language is a concatenation, and we need some english words to explain the fundamental concepts which can define concatenation). The same thing happens with mathematics. When mathematicians try to define the number 2 they're already using this concept, because the "idea" of two is already present in concepts such as dyadic relations, or english particles with two letters. So, you should focus on the theory that has more practical use and concision. Take intuitionism for example, although it has some very interesting points of view, it couldn't even build up classical analysis, so it isn't very usefull. Russell's logicism, although accepts the notion of universals such as relations and classes, derived all mathematics using only the logic of relations, so it's worth to pay attention to it. Be carefull with what people say about logicism, they tend to be exaggerated, he defined mathematics as logic and logic as mathematics, so his ideas didn't please mathematicians who liked to think of logic as some separated philosophical branch without very much use.

Have a nice day.

Ricardo
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  • Intuitionism did not *fail* to build up classical analysis, it accepted a limitation on infinity that made it contradict classical analysis. A lot of the structures that most concern classical analysis simply did not exist in intuitionist construction and so things like continuity come to lack meaning. Since all of the results of classical analysis ruled out by intuitionism either require things one cannot construct, or otherwise cannot be considered helpful, this is not a failure, it is an ontological position. –  Oct 13 '14 at 18:26
  • @jobermark E.g, whatever intuitionism construct, it's not classical analysis. – Ricardo Oct 13 '14 at 18:41
  • Just pointing out that when you talk about foundational principles, failure is relative. Does ZF *fail* to deal with the collection of all groups? Then does that mean traditional set theory 'cannot even' reach the accomplishments of intuitionism in abstract algebra? Of course not. –  Oct 13 '14 at 18:44
  • @jobermark I know, I agree with you. Excuse my poor choice of words. What I mean is that classical analysis is needed, and the intuitionist approach does not provide that. – Ricardo Oct 13 '14 at 18:49
  • No, not really, something that matches the testable part of classical analysis is needed. And both approaches provide that. Whether they should agree on the deeper, more philosophical level, that cannot get to the point of application, is really debatable. For instance, physicists use the 'delta' function, a continuous point function, which does not 'really exist' in classical analysis, but does in intuitionism. So what does 'necessary' mean? –  Oct 13 '14 at 18:51
  • @jobermark I don't like to think that way. The more philosophical level is the justification of all the corpus of knowledge. Of course, ZF and PM leads both to cardinal arithmetic, but the arithmetic of natural numbers has untraceable origins, unlike most concepts of analysis which ware based on philosophy. I like to think of analysis as the logic of relations: if we move on philosophy, we change the whole thing. – Ricardo Oct 13 '14 at 19:01
  • OK, but you miss the whole point of having a philosophy of mathematics, if shifting the philosophy is something so dangerous we cannot ever do it. To put it more polemically than I really believe, just for clarity: ZF is a patch job on the failed Platonism behind PM, that turns it into a Formalism, rather than a natural kind of thinking. And it is therefore not satisfying intellectually to people who want a real philosophy of mathematics. E.g. If physics needs a delta function, a Formalist one is cheating, you never know exactly what the side cases are without tons of work that no one does. –  Oct 13 '14 at 19:06
  • Look at something like Lakatos "Proofs and Refutations" which plays out a simple analog or worse, Feyerabend's "Against Method" which details historical examples, to see how hard it is to fix the basis of a bad theory. But bad is bad, and math at the moment is close to bad. –  Oct 13 '14 at 19:14
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I would claim that mathematics is the systematic exploration of idealization and human intuition. The objects studied are real only in an idealized sense, and the operations must obey idealized rules that approximate reality in narrow ways that minimize acceptance of external data.

So I would not claim that it is particularly about the integers, but your last statement fits my experience best.

The first situation is actual Platonism, the second is Formalism. These two approaches dominate the field in the sense that "Your average logician is a Platonist on weekdays and a Formalist on Sunday."

The third position is most clearly reflected by the project of Intuitionism, which tried to resolve the issues of Russel's paradox, etc., by questioning the natural force of negation and considering mathematics more a joint psychological endeavor that requires the investigation of our shared intuition, rather than a reflection of external or formal constructions.

Unfortunately, changing the meaning of mathematics requires reconstructing what is already known in another form, and such projects do not broadly capture the imagination of working mathematicians (though it makes better headway among those drawn to other computational disciplines.)

  • Wooa Wooa systematic? There is nothing systematic in **ANY** research and thinking. Systematic can be only **METHOD** -- the single tool of enlightenment which helps us to see facts. There are always facts outside any methods and that is where imagination is needed. – Asphir Dom Oct 13 '14 at 20:16
  • The notion of writing proofs and communicating them in certain notations is indeed a system. Outside of that, it is hard to see things as mathematics. I would contend (after Kuhn) that it is the attempt to be systematic, to keep a set of paradigms functioning that makes any research or thinking a science. So to the extent mathematics tries to remain a science, it is in fact systematic. –  Oct 13 '14 at 20:22
  • Imagination is still part of the system, we record our imaginings and compare them to others. –  Oct 13 '14 at 20:27
  • That what you described is not mathematics. It is a society. Order and organization is an **INNATE** property of mathematical objects. That does not give us right to be mistaken that mathematics is systematic on its own. Mathematics as a creation and exploration knows no system otherwise there will be nothing to discover. – Asphir Dom Oct 13 '14 at 20:30
  • Our mathematics is a social endeavor, with sociological wrapping. That wrapping could be different, but to imagine it can disappear completely is silly. Systematized as it is there remains an immense quantity to discover, so I don't get what you mean. –  Oct 13 '14 at 21:27
  • @AsphirDom Your very notion that mathematical objects have innate properties (much less telling what they have to be) prejudices the issue and forces a Platonic interpretation. But we know that such an interpretation leads right into Russel's Paradox. So why force things in a direction that fails. –  Oct 13 '14 at 21:31
  • I am at a loss to understand why Russell's paradox should undermine Platonism. Plato's idea forms are not ** extensions **, like sets are. – nwr Oct 14 '14 at 01:30
  • @NickR But sets are ideal forms. So is the notion of negation or error. And so is the notion of completeness or extension itself. And if you put them all together, one of them has to go. Also, I guess I should have said mathematical Platonism, which assumes our mathematical structures exist in the same sense as Platonic ideas. But I am pretty sure Plato would have agreed with that. –  Oct 14 '14 at 14:10
  • It is not at all clear that human mathematics resembles Plato's ideals. In the last 50 years it has become apparent that many of our theories are redundant. E.g., Galois Theory is just another way of formalizing the theory of analytic functions in the complex plane. Similarly, our theory of elliptic curves is just another articulation of complex analysis. This is all tied up with something called Langlands Program. Russell's paradox tells us that the concept of collectivization is not well-defined, and so is not part of Plato's vision. (continued...) – nwr Oct 14 '14 at 16:53
  • (... continued) I used to think that Plato's ideal world must contain itself and therefore was inherently inconsistent. To be fair, I still think it is inconsistent, but for other reasons. Similarly, Godel's incompleteness result does not necessarily imply that Plato's world is incomplete since it is not necessarily an axiomatisation. I would call myself a fuzzy Platonist. – nwr Oct 14 '14 at 16:55
  • (Context matters.) Whatever other notions of Platonism you might hold from a more completely philosophical stance, we are talking about philosophies of math here. So the definition of "Platonism" we mean is that mathematical structures have Platonic reality. A position that makes them real ideals outside the human process does, in fact, lead straight into Russel's paradox. And this is the force behind what @AsphirDom was asserting about their innate order. –  Oct 14 '14 at 19:03