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Kerrie

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mathematics is description of the reality of the universe

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what this language describes is another question.... sometimes people find out correlations with the external world only very much later than the formulation

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selfAdjoint

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Mathematics is the study of patterns. This is a recursive study because the relationship of two patterns itself makes a pattern.

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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.

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selfAdjoint

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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.

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Kerrie

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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...Originally posted by selfAdjoint

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.

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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?

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selfAdjoint

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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.Originally posted by sascha

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?

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.

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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.

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For some kids, Mathematics is a nightmare.

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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.

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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.

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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.....

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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.

cheers,

phoenix

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Jug

Mathematics is the science of describing exacting relationships.

"All things number and harmony." - Pythagoras

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NateTG

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I like to think of it as the study of abstraction, but the notions are essentially the same.Originally posted by selfAdjoint

Mathematics is the study of patterns. This is a recursive study because the relationship of two patterns itself makes a pattern.

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just out of curiosity, what is the point in defining terms like mathematics, art, philosophy...?

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but this thread is titled "what is mathematics" not "what is the nature of mathematics?"

having said that, i think my definition of math captures some of its nature.

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ah, interesting. I was thinking that the point of asking the question "what is mathematics?" is to debate the nature of mathematics; otherwise one can collect definitions anywhere. On the other hand, since in your own opinion defining mathematics is pointless, and that we should discuss its nature, why do you yourself give a definition (your post of 6 October)?

By the way, I do think your definition leaves out some rather essential features. Think of the basis of projective geometry ("Geometrie der Lage", Felix Klein and others) that contains all geometries and sets out on a purely qualitative path (i.e. with no formal definitions, only the intuitive idea on the point, the line, the plane, in space). By thinking infinity rigorously at every step, it gradually develops all that is needed, including the concept of counter-space, or imaginary points. In this geometry, infinity is never a special case. And numbers enter the scene in a natural way, not as with set theory, where you always have the problem whether the (postulated) entity called "set" is an entity of the same sort as the others which one wants to deal with. The result of the problem is Russell's antinomy.

The projective geometry line of mathematics has difficulties within your definition, which seems in other words a bit too narrow. But by definition, definitions are always too narrow, because they nail down the subject matter before it is explored. In this context: What do you think of my post of 29 September?

By the way, I do think your definition leaves out some rather essential features. Think of the basis of projective geometry ("Geometrie der Lage", Felix Klein and others) that contains all geometries and sets out on a purely qualitative path (i.e. with no formal definitions, only the intuitive idea on the point, the line, the plane, in space). By thinking infinity rigorously at every step, it gradually develops all that is needed, including the concept of counter-space, or imaginary points. In this geometry, infinity is never a special case. And numbers enter the scene in a natural way, not as with set theory, where you always have the problem whether the (postulated) entity called "set" is an entity of the same sort as the others which one wants to deal with. The result of the problem is Russell's antinomy.

The projective geometry line of mathematics has difficulties within your definition, which seems in other words a bit too narrow. But by definition, definitions are always too narrow, because they nail down the subject matter before it is explored. In this context: What do you think of my post of 29 September?

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which 9/29 post?

i'll guess this one:

however, to me, this is like defining a chair by saying "a chair is an object." in other words, yes it is a language but just saying that leaves about as much to the imagination as "a chair is an object" leaves about a chair.

whether or not math is more formal than say, english or latin, is disputable. when i look at a grammar book, it all looks pretty formal with all its rules to me. but this isn't really a central issue. i suppose the "grammatically correct" statements in mathese would be theorems and proofs.

what is the point? to kill time i guess.

i'm not seeing how projective geometry is not in my definition of math. "with no formal definitions, only the intuitive idea on the point, the line, the plane, in space." in my number 3, i said that some terms are not defined. one could call this lack of definitions an intuitive/qualitative domain. in fact, geometry and set theory were two things i had in mind when i said that not everything has a definition, eg sets and points. if you use the word "infinity" then you're doing a special case of #3 in my definition.

the projective geometry i'm familiar with is not at all just qualitative and it has formal defnitions.

i'll guess this one:

i looked at the definition of the word language on dictionary.com and mathematics seems to be a language.

what this language describes is another question.... sometimes people find out correlations with the external world only very much later than the formulation

however, to me, this is like defining a chair by saying "a chair is an object." in other words, yes it is a language but just saying that leaves about as much to the imagination as "a chair is an object" leaves about a chair.

whether or not math is more formal than say, english or latin, is disputable. when i look at a grammar book, it all looks pretty formal with all its rules to me. but this isn't really a central issue. i suppose the "grammatically correct" statements in mathese would be theorems and proofs.

what is the point? to kill time i guess.

i'm not seeing how projective geometry is not in my definition of math. "with no formal definitions, only the intuitive idea on the point, the line, the plane, in space." in my number 3, i said that some terms are not defined. one could call this lack of definitions an intuitive/qualitative domain. in fact, geometry and set theory were two things i had in mind when i said that not everything has a definition, eg sets and points. if you use the word "infinity" then you're doing a special case of #3 in my definition.

the projective geometry i'm familiar with is not at all just qualitative and it has formal defnitions.

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In fact I meant my post after the one you quoted, the one after selfAdjoint. There I say "Geometry has been engulfed by algebraic analysis", which corresponds to what you tell me about the projective geometry you are familiar with. This is just my point: the universality of the subject matter (as possible in a purely qualitative approach, thinking in geometrical way of thinking, in continuous terms) has been lost by subjecting it to the gesture of algebraization, thinking in discontinuous terms. This is the fashion nowadays, and mathematicians have not yet grasped what they have lost. As little as mainstream philosophers, by the way, who also resorted to thinking in discontinuous terms.

My saying mathematics is a language is not meant as a definition, but as a clarification for taking off some of the false gloss which some put onto it, believing mathematics is more than a language (e.g. something that warrants per se correctness in dealing with material reality). It is possible that #3 in your definition covers intuitively what I mean. But I think the distinction of continuous versus discontinuous approach still needs to be more explicit for expressing fully what I mean. This is indeed not discussed very often. The last I remember is an article by the Bourbaki group, many decades back. You are right in your post of 6 October that everything deeply considered becomes a branch of philosophy. But this is valid even today.

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what do you mean by contiuous?There I say "Geometry has been engulfed by algebraic analysis", which corresponds to what you tell me about the projective geometry you are familiar with. This is just my point: the universality of the subject matter (as possible in a purely qualitative approach, thinking in geometrical way of thinking, in continuous terms) has been lost by subjecting it to the gesture of algebraization, thinking in discontinuous terms. This is the fashion nowadays, and mathematicians have not yet grasped what they have lost.

i don't think mathematicians are not "thinking in geometrical way," in general, when they think about projective geometry. it's not like they state a bunch of equations without ever refering to geometric objects. and i'll bet that the geometric way of thinking inpsires the direction the algebra is taken in. i'll also bet that directions in algebra can give new directions in the geometry. i think to totally ignore either side of it would be to lose something essential. i'm not seeing what's wrong with codifying geometric intuitions in algebraic form.

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