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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.

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    For teh frist one, see Platonism; for the second see Formaism and for the third onesee Intuitionism. In general, see Philosophy of Mathematics. – Mauro ALLEGRANZA Oct 12 '14 at 13:38
  • And there are also moder recent issues : see Naturalism and Indispensability Arguments. – 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. – Nick Oct 13 '14 at 1:14
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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.

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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. – user9166 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. – user9166 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? – user9166 Oct 13 '14 at 18:51
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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.)

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  • 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. – user9166 Oct 13 '14 at 20:22
  • Imagination is still part of the system, we record our imaginings and compare them to others. – user9166 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. – user9166 Oct 13 '14 at 21:27

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