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What is Mathematics?

  1. Sep 28, 2003 #1
    What is mathematics? I like to think of mathematics as the language in which nature speaks... What is mathematics to you?
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  3. Sep 28, 2003 #2


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    mathematics is description of the reality of the universe
  4. Sep 29, 2003 #3
    mathematics is a language -- in contrast to others a thoroughly formalized one

    what this language describes is another question.... sometimes people find out correlations with the external world only very much later than the formulation
  5. Sep 29, 2003 #4


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    Only two replies and already a division of the house. Let me add another idea which doesn't square with either of the first two.

    Mathematics is the study of patterns. This is a recursive study because the relationship of two patterns itself makes a pattern.
  6. Sep 29, 2003 #5
    selfAdjoint, a division of the house can be inspiring. What you say -- "mathematics is the study of patterns" -- is closer to what I say than you may believe, because I had been thinking about posting that this language is about patterns of the mind (not describing them, but depicting / revealing them). Note that the recursiveness you mention is only formal, it does not cover content: the pattern resulting from relating two patterns is not the same pattern.

    Something that bothers me more than that is one pattern that has been overgrowing and obscuring the rest: the gesture of algebraization, thinking in discontinuous terms -- as opposed to the geometrical way of thinking, in continuous terms. Geometry has been engulfed by algebraic analysis (while it makes quite a difference whether you understand e.g. a function as a curve or as a formula). Strangely enough, this difference is extremely rarely even discussed -- but has an enormous impact on the way thinking as such, as well as thinking about the world, are being understood.
  7. Sep 29, 2003 #6


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    Sasha, in grad school I discovered that different mathematicians have different talents. It appears that the algebraic talent, doing things with abstract counters in your head, is more widely distributed than the geometric one, manipulating pictures in your head.

    By which I don't mean that either algebra or geometry is those simple qualities, but that those talents are what you need to succeed in algebric or geometrical fields.
  8. Sep 29, 2003 #7


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    so could the definition of mathematics be based on personal perspective of it? i see math as a description of our world around us, in the geometric sense, as basic geometry comes super easy to me...i suppose i can see why people call it a language-a universal one, but to me, language is a man made structure of communication rather...
  9. Sep 29, 2003 #8
    By language I mean its very principle: being a structure of several signs, interrelated according to a basic idea (axiom, choice, desire, principle, etc.) -- as opposed to the very principle of thinking, which is to integrate and find the common basis of diversities. These two principles are not man-made; only this or that language, and this or that thought, are man-made.

    Sure, a talent for this or that is not distributed evenly. But here we are seeking, if I understand this thread correctly, the systematic reason and structure of math, not some contingent aspects. Or not?
  10. Sep 30, 2003 #9


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    Well I see mathematics as a human enterprise, more or less inseparable from its sociology, like political science. A platonist would say the math is "out there" somewhere, but I think it exists in people's heads.

    The reason it seems "out there" IMO, as I've posted before, is that the ideas that are accepted into the canon - published in journals and so on - have to be "well defined", meaning that their defining properties are clearly stated and related to each other (usually with an axiom system) so that it is no longer possible for informed people to disagree on their natures. This gives them a mental "solidity" similar to that of perceived rocks and chairs.
  11. Sep 30, 2003 #10


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    I veer between Platonic:

    Mathematics is the language in which the laws of the universe is written.

    And formalism:

    Mathematics is the language in which the laws of the universe is read.
  12. Sep 30, 2003 #11
    Maybe the point is less an 'either–or' (of e.g. the Platonist vs. formalist approach), but of realizing that the aspects and interconnections between laws and their appearances on all levels are precisely a question of how we approach totality, 'the whole thing'. I advocate thus an 'as-well-as' overall view. This does not mean to relativize blindly, but to relate any (assumptional, presuppositional) approach to its (logical, predicative) effect and gradually finding an ever more complete grasp of grasping -- i.e. not only seeing the laws or their effect, the 'writing' or the 'reading', but both. AFAICS, there are presently four types of approach to the philosophy of mathematics, of which none is strictly conclusive: Platonist, conceptualist, formalist, and intuitionist. My aim is more to understand the reasons (relation between assumptions / presuppositions and their logical effect) behind these approaches, rather than to attach myself to one of them.
  13. Oct 1, 2003 #12


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    For some kids, Mathematics is a nightmare.
  14. Oct 2, 2003 #13
    Mathematics is a nightmare only when taught in ways that leave no space for the creative quests that arise when hitting upon the structures of well organized thinking. Of course, in an epoch like ours -- in which everything is being formalized and instrumentalized, making it lifeless, sacrificing the overview for getting lost in details -- math can indeed be made into a nightmare. I remember how reading George Polya aroused my interest after having felt for a long time that math is a real drag. In France there was an interesting book "Echec et maths" (a double word game: failure / chess and checkmated / maths), showing how kids are pushed into falling though, as an effect of the mentioned 'pedagogical' ideas.
    Last edited: Oct 2, 2003
  15. Oct 2, 2003 #14


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    I was unlucky enough to have been taught in a system that brings nightmare (oh yes the real ones). My first school was in Geneva and when I was about 7 or 8, I went back to Burma. Then comes the shock: everything turned upside down from the moment I first entered the school. I hated school ever since then, and especially math. But luckily the story didn't end there, and gradually I came to love physics and math.

    To get back to the topic; I imagine mathematics is a language that describes what our oral languages lack to describe. Quantitively. But far from being universal because I believe there are many flaws in it. Mathematics doesn't describe the principles and laws but merely a language that we use to understand the description of the universe. If mathematics is a language universal and can describe every principles in the universe, we would have found the grand unified theory itself.
  16. Oct 3, 2003 #15
    There are probably only contingent pragmatic reasons why mathematics is usually subdivided into a "pure" and an "applied" sector. The question then is, in which one does one situate oneself, maybe even forgetting about the other one. This might concern your characterization of mathematics as something quantitative. This may be reasonable to some extent for applied math, but at least pure mathematics ought to be more open than that, otherwise it will gradually self-limit itself, become one-eyed, etc..

    In this problem, I would like to mention a point that Hegel has demonstrated quite clearly and which should not be forgotten: All quantities require, for being formulated, a qualitative decison. Any quantity implies a qualitative foundation. It cannot be defined without that. Of course one can forget about this foundation. But that is quite another question. Then one will face the consequences in inherent limits of the resulting systems -- which is especially fateful when operating in a discipline which should be free of biases for remaining on track.

    It is interesting that in a former thread here on mathematics (now in the Archive), the participants had more or less agreed on the idea that mathematics is based on the numbers -- and then were compelled to argue endlessly about the resulting problems. "It is all in the head", the problems are all self-made.....
  17. Oct 6, 2003 #16
    different mathematicians will give different definitions.

    here's one way:
    mathematics is a branch of knowledge characterized by the following:
    1. rules of logic and deduction are studied and/or adopted.
    2. what is considered a proof is studied and/or agreed upon.
    3. terms are used with or without having a definition. the words without definitions are kept to a minimum but that they are undefined remains recognized.
    4. assumptions are studied. some are agreed upon.
    5. theorems are written which combine steps 1, 2, 3, and 4.
    6. consequences of theorems (using 1-4) are written.
    7. attempts at generalizing the theorem are made.
    8. conjectures are studied; one looks for them to be proven as either a theorem, not a theorem, or undecideable.

    within this greater definition, you can have a wild variety of the particulars when it comes to steps 3 and 4.

    sometimes, a particular case of mathematics will in some way resemble reality but that is NOT the defining aspect of mathematics unless by "reality" you mean more than the physical/observable. keep in mind such fields as metamathematics and category theory.

    whenever one attempts to define mathematics, it is often the case that a field within mathematics would be left out by that definition. the average definition would not include category theory, metamathematics, and arithmetic; i believe that this definition leaves no field out.

    however, in its generality, other things not commonly considered mathematics are drawn in such as philosophy and science, but this appears to be a part of a larger suspicion that many branches of knowledge resemble each other at the abstract level. one can probably invent a similar definition of philosophy and then consider mathematics a branch of it. once, everything was considered a branch of philosophy.

  18. Oct 26, 2003 #17


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    What is Mathematics

    Mathematics is the science of describing exacting relationships.

    "All things number and harmony." - Pythagoras
  19. Oct 31, 2003 #18


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    I like to think of it as the study of abstraction, but the notions are essentially the same.
  20. Oct 31, 2003 #19
    just out of curiosity, what is the point in defining terms like mathematics, art, philosophy...?
  21. Nov 1, 2003 #20
    Phoenixthoth, why does for you asking a question about the nature of a field or discipline amount to defining a term?
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