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I did not ask this question before scaring of down-voting but could not stop the curiosity and cannot find the answer by searching the web. In physics we are looking for say smallest mass or particle, many many years ago it was electron may be, sometime ago it became quarks and we are still looking for it.Is there anything equivalent in mathematics? I know there are many unsolved mathematical, we can search in the web and find many, but that does not answer the question: are we looking for something fundamental? Are we looking for something fundamental finding which will solve many present problems?

Sorry if the question was not clear, but the answer provided by @jack and edited by@celtschk, gave me whatever I was looking for. As jack said the concept I was asking is not an active research now and that exactly I was looking for.

Creator
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  • Depends on who you ask. Wolfram would say yes, and will say he has found it (cellular automata). There was a time in the past when there was a concerted effort to prove axiomatic systems consistent. But Godel put an end to that. At this point there's no real fundamental goal like that. But there are a few landmark conjectures that would revolutionize mathematics. The Riemann Hypothesis, for example. – Gregory Grant May 02 '15 at 19:32
  • The Riemann Hypothesis, clear and simple. – Hasan Saad May 02 '15 at 19:32
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    No. Mathematics searches are many fold and variate, and among other things it lays foundations for physicists to do their things. – Timbuc May 02 '15 at 19:32
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    There's no goal for all of mathematics, because mathematicians have very different interests. However, there are ''grand programs" and big open problems that have attracted a large amount of interest. – Jair Taylor May 02 '15 at 19:32
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    @HasanSaad, Hmm, what good is the Riemann Hypothesis unless it helps prove P != NP? :) – Jair Taylor May 02 '15 at 19:35
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    It's good for its own sake, just because proving things is fun. Now, I know I'm still a beginner and that I don't really understnad it, but seeing that it relates to analysis and to complex numbers, I plan to solve it one day. That was the thing I most desired when I started learning maths. – Hasan Saad May 02 '15 at 19:37
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    @HasanSaad Even if this was true, the the goal would not be the RH, but it would be the understanding to (the distribution of) primes. – Pavel Čoupek May 02 '15 at 19:37
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    There is no one fundamental goal, per you title, but there are lots of questions that are linked. I'd suggest your view of physics history is naive, and thus the parallel has problems. – Thomas Andrews May 02 '15 at 19:37
  • Well, you might be right. RH would then be a secondary goal, but still one must look at it as the goal before one can look deeper. You can't look at the sea if you have ten thousand gates before it you know :p – Hasan Saad May 02 '15 at 19:37
  • Does this qualify? – celtschk May 02 '15 at 20:03
  • World domination. – Gerry Myerson May 03 '15 at 06:42

4 Answers4

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I think this is an excellent question, and I would hope no one would down vote it.

There was a time when mathematics had a similar goal. There were several issues found in the late 19th century which demonstrated that a naive approach to mathematics led to contradictions. For example, the set of all sets that do not contain themselves is a contradictory object. Does it contain itself? If so, it cannot contain itself. This is Russell's Paradox. This and other things showed that mathematics needed to be founded on a precise foundation that laid out exactly what each mathematical statement meant in full rigour. Such languages are called formal languages.

The effort to resolve these issues was known in the early 20th century as the Foundational Crisis of mathematics. It was hoped that a foundation could be found that covered all that ought to be covered in mathematics, and that was known to be consistent. In particular, David Hilbert published some problems he thought to be important for the development of mathematics into the 20th century, and the second of these problems was part of the ideal foundation: to prove that the axioms of arithmetic were consistent.

It turned out that the axioms of arithmetic cannot be proven consistent in first order logic without assuming the consistency of some other first order system of enough complexity to kind of make a consistency proof not mean much. This is Gödel's Incompleteness Theorem, and it is of course precise, but I just summarized it. Then Turing and Church showed that one cannot solve the question of whether or not an arbitrary computation ever reaches its goal, and this has deep implications for what can be done with foundations. After these discoveries, it was sort of de facto concluded in the mathematical community that an ideal foundation was impossible. Hence there is not much of an active search for one right now.

It should, however, be noted that Kurt Gödel himself recognized that his proof only ruled out one way of having such a foundation. Gödel never said that an ideal foundation for mathematics was impossible, and it seems that he may have even believed the contrary. Nonetheless, mainstream mathematics is not concerned with finding an ideal foundation at the present time.

celtschk
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jack
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    The question did not ask if there is a fundamental foundational goal in math, so this does not appear to answer the question. – Bill Dubuque May 02 '15 at 20:21
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    The analogy to fundamental particles seems to be getting at if there is an analogue in mathematics to the search for the Unified Theory. After all, finding the smallest particle and finding the Unified Theory are likely very intertwined future events. The key aspect of the Unified Theory is that it provides a context in which all physics is understood. Thus an analogue in mathematics would have to be a foundation providing a context in which all math is understood. That is the sort of foundation that was desired in the early 20th century. – jack May 02 '15 at 20:30
  • I was thinking more along the lines of projects like the Classification of Finite Simple Groups, or the Langland's program. Certainly, the efforts to understand the foundations of mathematics is one such effort - it was a large program across many mathematicians trying to come to an understanding, and which has strong repercussions today. – Thomas Andrews May 02 '15 at 20:52
  • @Thomas Andrews, I think a main point with the analogy to fundamental particles is the property of underlying literally all of physics (by relation to the UT). There are projects like you say in physics, like classifying all possible characterizations of stars, although these projects tend to be less defined pursuits. However, the UT is unique in its universal quality of underlying everything in the field, and it is the UT that is most associated with the notion of finding smallest particles. Only a mathematical foundation could be analogous to the UT with respect to that universal property. – jack May 02 '15 at 20:59
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In my opinion, mathematics is an art, not a science.

So, this question is as relevant as asking "Is there a fundamental goal of painting, or literature, or movies?"

There are many goals, some personal, some societal, some financial, some, some, some.

I know I love doing math (the limited amount that I can do), and hope to continue doing it for quite a long time.

marty cohen
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I don't think your characterization of "the" fundamental goals of physics is accurate. At any given time, there are unsolved problems in physics, and there are unsolved problems in mathematics, and these problems maybe catch the interest of a lot of people, or maybe a smaller number of people, or maybe just a few people.

I think physics and mathematics are actually quite similar: we explore, we try to discover new true things, we try to demonstrate their truth rigorously using appropriate methods (experimental method in physics; logical proof in mathematics).

If there is a difference in this regard between mathematics and physics, it is that the number of interesting and tractable problems in physics is rather more tightly constrained than in mathematics, because a physics problem must comport with physical reality. Mathematicians are much freer in that regard, although they still must comport with logical reality.

Lee Mosher
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  • I think the idea of the question is whether or not mathematics has a goal similar to the search for a Unified Theory. Physicists have many different interests, but there is a goal to find a theory that would literally explain all of physics, at least in principal. It does make sense to ask for an analogous pursuit in mathematics. – jack May 02 '15 at 19:59
  • @Jack: Part of what I'm saying is that I disagree that that goal exists, even in principal. I'll grant you that that goal gets a lot of press, though. – Lee Mosher May 02 '15 at 20:00
  • Although I see little reason to discount the validity of the pursuit of the Unified Theory, I can accept that you do not see it as a valid pursuit. However, it is a matter of fact that it is a goal for a large part of the physics community. – jack May 02 '15 at 20:02
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Mathematics is the study of consequences. For instance, what is the consequence of Newton's gravitational inverse-square law? Using mathematics, one works out the theory of celestial mechanics, and more broadly, classical mechanics.

What are the consequences of Einsten's observation that the speed of light in a vacuum is independent of the reference frame? Mathematics shows us that special relativity must be true.

Ultimately, the place of mathematics in the rest of the world, is that we take the observations from other field of study (physics, chemistry, biology, medicine, etc..) and work out the logical consequences of those observations.

The Hawking/Penrose prediction of black holes as a consequence of Einstein's theory of general relativity is one of the great achievements of mathematics, for instance.

treble
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  • Read the question, he's not talking about the big goal of mathematics, or a definition of mathematics. He's looking for parallels with the ideas "electron" and "quark" in physics. – Thomas Andrews May 02 '15 at 19:40
  • I have read the question, and the first sentence of my response is a concise answer to it. – treble May 02 '15 at 19:42
  • Sorry, it is not. You are answering the title, without giving any hint that you understand he is asking about specific programs of efforts. (An equivalent in mathematics was the classifcation of finite groups, for example.) – Thomas Andrews May 02 '15 at 19:46
  • `You are answering the title..' is the title not the question? Do you think that OP has asked more than one question, and so it has upset you that I have only answered the titular one? – treble May 02 '15 at 23:33