@hemantah: your starting point is totally wrong. Mathematicians are trying to sell themselves to the outside world for sociological reasons, but the development of math is more pure than it ever was! Think that the concept of pure math did not appear before Hilbert say...

In my opinion, the situation is as follows: before, the scientific, economic and technological development of the society was so low that their major theoretical problems and the elimination of the restrictions in the field of sciences and technology did not require deep knowledge of mathematics or the use of sophisticated models and complex tools to help in their development. At that time, mathematicians spend time to deepen their knowledge of the mathematic sciences and the study of physics, philosophy, astronomy, among other sciences.

Once the scientific and technological development of the society was growing and becoming increasingly more complex, the solution of major problems and the study of phenomena associated with economic, scientific and technological development of the same were requiring deeper mathematical knowledge in order to be able to apply more and more sophisticated models and tools offered by the mathematics to find solutions to those problems, including those associated with science itself. The pressure on the mathematicians for the development of new and more sophisticated models and tools to be used in the search for the best solutions that limited scientific and technological development of the society made mathematicians focused more on the application of advances in mathematics than in former times.

However, the search for these new models and tools require both deep theoretical knowledge and the development of the mathematical sciences from a theoretical point of view. Of course it is more difficult to find now new and more sophisticated mathematical models and tools to continue the development of the mathematical sciences from a theoretical point of view and is most frequent to see application of mathematical models and tools already existing in the solution of the problems arising in the field of science, economic and technology. Perhaps, this situation give the wrong impression that now mathematicians are focused more and more to develop applied mathematics that to the development of pure mathematics. At the same time, economic crisis, limited resources available in many countries for the development of l sciences, difficulties in the preparation of high qualified professionals in the fields of sciences and particularly in the mathematical sciences in many countries, among others are forcing mathematicians to focused in the application of mathematics in the solution of real problems affecting the society than in the search for new theoretical models and tools in the field of pure mathematical science.

However, the main principles based on which the mathematic sciences have been developed are the same and have not suffered significant changes in the past hundreds of years.

@hemantah: your starting point is totally wrong. Mathematicians are trying to sell themselves to the outside world for sociological reasons, but the development of math is more pure than it ever was! Think that the concept of pure math did not appear before Hilbert say...

@Patric: Development of mathematics is according to you more pure than it ever was, whereas mathematicians are trying to sell themselves to the outside world for sociological reasons.

What sort of sociological reasons? Was mathematics less pure earlier?

For example Ramanujan developed mathematical formalisms without bothering to think whether his findings would be applicable in certain fields. That was the philosophy of mathematics in those days. Now of course, some of Ramanujan's findings are actually used in computational mathematics. But Ramanujan discovered those things purely as mathematics.

Currently, the applications are perhaps given more importance. That way, the philosophy of mathematics is perhaps changing a bit. That is what I think.

@George: You have actually supported that currently mathematics has started to become a slave of application oriented fields. In other words, you have supported that the original philosophy of mathematics is no longer there.

Hemanta. Well this is not my conclusion. In my reply I said: Perhaps, this situation give the wrong impression that now mathematicians are focused more and more to develop applied mathematics that to the development of pure mathematics.

@Jorge: So according to you, the philosophy has not really changed. Actually, the people of mathematics are not really in the process of changing the philosophy. It is the users who are doing so. That is what I feel; I may be wrong.

Personally I think the situation in mathematics has not changed. Also in former times mathematicians tried to solve practical problems as well as pure mathematical ones. An example for applied mathematics in ancient times would be the development of basic geometry in old Egypt for land surveying after the periodical floodenings of the nile.

An example for "pure" mathematics would be the discovery of and research on prime number in ancient Greece.

And this goes on and on throughout history:

Fibbonaccis series for describing the population evolution of rabbits (applied maths) vs. Al-Battani who introduced the concept of the number "0" ("pure" maths) in medieval times.

Newton's calculus for solving physical problems (applied maths) vs. Fermat's number theory ("pure" maths) in the 17th century.

In the 20th century we have, for example, Allan Turing with his work about computers (applied maths) vs. Kurt Goedel with his incompleteness theorem ("pure" mathematics).

So, I think it has always been the same, but I fail to see what difference does it make whether someone applies mathematics to a certain problem or whether one solves problem just for the sake of solving it (which might also lead to some practical applications in the future). This does not affect the maths itself - you could also say: "there is no better or nor worse mathematics, it is beautifull in any case."

@Johannes: Yes, not all mathematical results have been just theory oriented. Calculus as you have mentioned, was discovered while trying to explain gravitational laws. Indeed, what I meant was a bit different,

For example, take some statistical problem. Earlier, before the advent of software and fast computers, people consulted the available theory for analysis. Now a days, the users do not take care to see whether in some particular case a given formula is applicable or not. Mostly, the users just use the available software. This kind of use has led to certain types of data analysis based on which certain mathematical formalisms have come up. It is this kind of mathematics I was referring to. This kind of computer dependent mathematics has perhaps been slowly changing the philosophy of mathematics.

@Hemanta Baruah:

Sorry, but I don't see why the use of computers should change the philosophical issues of mathematics.

What you describe is just "bad science" - taking anything without thinking about it first, is just sloppy working, but that does not have any influence on the maths itself.

An example: I was once helping out a colleague from the theoretical department with his diploma thesis by supplying him with some data he needed. He did it the right way - he used a computer programm (COMSOL) for his simulations, but still he checked every single equation in the programm toolkit before he used them.

But also here things haven't changed that much - in former times people used tables (for example, for integrals, logarithms,...) to look up some stuff, nowadays it is computers. However, this should not be taken lighthearted (without thinking) :)

Maybe I still don't get your point, so let me ask a question: what exactly would you define as "philosophical aspects of mathematics"?

@ Johannes Gruenwald: Suppose you have observed a physical fact. Then you have gone forward to explain that logically. That would lead to a mathematical formalism. What I mean is that logic should be followed by mathematics, it should not be the other way around.

Sir Issac observed a physical fact that led to the laws of falling bodies. Here was an example how mathematics followed logic.

Similarly, long back people observed that

(1) sum of two integers is an integer again,

(2) for example, 2 + (5 + 7) = (2 + 5) + 7,

(3) 0 added to an integer is equal to that integer again, and

(4) for example, 5 + (-5) = 0.

These were observed facts. Then these facts have led to four axioms that define the Group of Integers. That is how mathematics has been growing, following logic to explain observable facts.

Axioms do not fall from the blue. But suppose someone suddenly comes up with an absurd idea of defining a triangular quadrilateral. Based on that kind of a nonsense, if an axiom is stated based on which some weird mathematics is done, then those who do that would have to force logic in their kind of mathematics. This is what I am objecting. Logic must not be forced upon some mathematics done based on illogical axioms.

In the name of data analysis, certain mathematical formalisms have started to come up of late. Not all of them are logical perhaps. But logic is forced upon that kind of mathematics, which goes against the philosophy of mathematics.

Ah, now I get your point - sorry that it took me so long.

But I still see things different. Mathematics cannot follow logic or vice versa, because maths is logic - so, if some operation is correct (in a mathematical sense) then it is automatically logical.

Unfortunately your examples (1) - (4) are not necessarily facts, they are rather based on the group properties of integer numbers.

Regarding the data analysis formalisms I cannot contribute something meaningfull to that statement because it is not my field of expertise, but your comment suggests that you know of at least one of those techniques which is not logical. Hence it cannot be mathematically correct. If this is the case then I would publish your findings, because it will help other researchers to avoid mistakes in the algorithms they use.

@Hemantah: by sociological reasons I mean the trivial need to be funded. If you really mean to talk about philosophy I think the parangon of abstraction and rigor was reached in the last century with the Bourbaki group and the Bourbaki treatises. I dont think the standards of rigor and abstraction have lowered down since. Thus the structuralist ideas are still there. The difference in practice might be more use of computers in research but when comes the time of writing a formal paper the language remains the same...Cf Mochizuki claimed proof of ABC!

@Johannes Gruenwald: There are too many examples I have come across. Take for example the simple case of finding the coefficient of determination r^2, where r is the simple linear correlation coefficient between two variables X and Y. Before finding out r^2, one has to see whether the relationship between X and Y is at least approximately linear. Indeed, for a nonlinear relationship, computation of r and therefore of r^2 does not make much sense. However, now a days people use software without bothering to see whether the relationship is linear or not, and declare the value of r^2 as the coefficient of determination.

I have cited just one example. There are far too many such examples. In fact, the multivariate extension of this, R^2, is used in all sorts of nonlinear cases in Demography for example. Based on this kind of wrong use of R^2, too many conclusions can be seen to have been taken in countless articles.

@ Patric Sole: The fraternity of mathematics is not to be blamed, some of the users are. Abstractism has not changed; for wrong use of a formula, the formula cannot be held responsible. What I mean is that a software is not to be blamed when someone misuses it.

The general philosophical and scientific outlook in the nineteenth century tended toward the empirical. Platonistic aspects of rationalistic theories of mathematics were rapidly losing support. Especially the once highly praised faculty of rational intuition of ideas was regarded with suspicion. Thus it became a challenge to formulate a philosophical theory of mathematics that was free of platonistic elements. In the first decades of the twentieth century, three non-platonistic accounts of mathematics were developed: logicism, formalism, and intuitionism. There emerged in the beginning of the twentieth century also a fourth program: predicativism. Due to contingent historical circumstances, its true potential was not brought out until the 1960s. However, it amply deserves a place beside the three traditional schools.

In the twentieth century, research in the philosophy of mathematics revolved mostly around the nature of mathematical objects, the fundamental laws that govern them, and how we acquire mathematical knowledge about them. These are foundational concerns that are intimately connected with traditional metaphysical and epistemological questions.

In the second half of the twentieth century, research in the philosophy of science to a significant extent moved away from foundational concerns. Instead, philosophical questions relating to the growth of scientific knowledge and of scientific understanding became more central. As early as the 1970s, there were voices that argued that a similar shift of attention should take place in the philosophy of mathematics (Lakatos 1976).

For some decades, such sentiments remained restricted to a somewhat marginal school of thought in the philosophy of mathematics. However, in recent years the opposition between this new movement and mainstream philosophy of mathematics is softening. Philosophical questions relating to mathematical practice, the evolution of mathematical theories, and mathematical explanation and understanding have become more prominent, and have been related to more traditional questions from the philosophy of mathematics (Mancosu 2008). This trend will doubtlessly continue in the years to come.

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Philosophy of Mathematics

Horsten, Leon, "Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2012 Edition), Edward N. Zalta (ed.).

http://plato.stanford.edu/entries/philosophy-mathematics/

Lakatos, I., 1976. Proofs and Refutations, New York: Cambridge University Press.

Mancosu, P., 2008. The Philosophy of Mathematical Practice, Oxford: Oxford University Press.