math=a biologist's worst nightmare
Every person has taken math in school some time in their life. BUT, why don’t we have a clear definition of Mathematics? What on earth are we studying? What kind of education puts you through 12 years of math and doesn't even explain to you what it is? Its funny/sad how we can't even define it in even a slightly coherent way. Which tells us a lot about how intelligent we and our teachers are.
Doing a google search on "what math is" doesn't really help either.
So I attempted to answer the question "what is math", and also since it is very close to science, I tried to explain "what is science", and compared them.
what is math
This forum is filled with people who know math and science, so I ask you to please view the video/article and criticize it as much as you can! your input is really valuable, thank you.
You are assuming that every term in use must have a precise definition, but this is not obviously true (except possibly to Socrates and Plato but they were evidently wrong, try defining the term "tall building"). All that is really required is that you recognise an examplum when you see one. You cannot define "dog" but you learn to recognise one when you see it.
If you insist on a definition; you could define Mathematics as the study of pattern and symmetry (in a formalised setting ....). But that won't really help you recognise mathematics when you see it without an awful lot of context.
Also your post is not a question but click bait. You think you already know something of the answer and are posting here to boost your site's number of hits. Confirmed by the fact that you have not logged into MHF since posting your click bait, which of course makes you a spammer (according to the definition I have here).
Also², your Einstein quote is an example of someone eminent in one field talking crap about another, Mathematics is not "independent of experience", it may be abstracted from experience but that is not the same thing as independence of/from experience. Which also nails the claim that mathematical propositions are a priori. If the space of experience did not appear to be a flat 3-manifold we would not have elementary geometry in the form that we do (we still might arrive at it as a limit in some sense derived from the space experience but that would be another story).
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First of all, thank you for your input.
Sooo, lets say I am driving traffic into my website, that's not necessarily bad, I am not selling anything on the website, nor do I have ads in it, and the content is supposed to be educational.. I haven't been in the forum since I posted because school started and I got distracted, but here I am! taking notes on people's input..
A) Can you explain why you can't have a precise definition? why is a "dog" undefinable? I mean just the fact that you can recognize a dog, doesn't that mean there is a definition of a dog, at least in your subconscious mind? I am confused, please explain this point to me.
B) and why is math just patterns and symmetry? what about everything else? like numbers? do you have a reason for that?
C) and finally, the quotes from different thinkers are just used for demonstration, it doesn't mean "if Plato/Einstein said it, it must be true!". Sorry if it seemed like that, I didn't intend it to be like that, the explanations after the quote are supposed to show why what I said is true/false...
Would being able to recognise an instance of "dog" have been sufficient to convince Plato's Socrates that you knew the definition of "dog"? Do you think everyone would agree that this entity you recognise as a "dog" is in fact a "dog". The term "dog" is fuzzy at the edges and if you draw a boundary it is essentially arbitary and you will not get universal agreement with where you have placed it. So I would contend that there is no definition of "dog" that you can give, even if you can recognise an instance of "dog" when you see one. You learn from experence what the term "dog" refers to (most of the time) but there is no definition because a definition is precise but here the concept is not.
Viewed in a certain light "number" may be related to pattern and symmetry. In fact number can be construed as a structure with translational symmetry. So when viewed in this light it is part of the subject matter of mathematics. There are other ways in which "number" displays symmetry and pattern ... But note the concepts "pattern" and "symmetry" are fuzzy just like that of "dog".B) and why is math just patterns and symmetry? what about everything else? like numbers? do you have a reason for that?
Confining maths to number and those who do not hate it is being too restrictive. There was a joke in the department where I was an undergraduate that the only numbers seen by maths students were those of the exam questions.
Unless the discussion is about what a particular author thought and the quote is to illustrate it, the use of quotes like this is an indirect appeal to authority.C) and finally, the quotes from different thinkers are just used for demonstration, it doesn't mean "if Plato/Einstein said it, it must be true!". Sorry if it seemed like that, I didn't intend it to be like that, the explanations after the quote are supposed to show why what I said is true/false...
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I like to think of it this way:
In every "study" we deal with specific objects and the relationships between them (In physics, the objects might be "mass" and "energy" or "distance" and "time", in chemistry, "elements" and "compounds", in history, "people" and "cultures"). In mathematic the objects we study are "relationships" in the abstract.
(One of the most general, and so abstract, fields of mathematics is "category theory" in which we look at the "category of sets", "category of groups", "category of topological spaces", etc. The "objects" in such categories are sets, groups, topological spaces, etc. while the "morphism" are function, homomorphism, continuous functions, etc. from one of those objects to another. One of the remarkable results of "category theory" is that it is possible to do away with the "objects" entirely and define the category entirely in terms of the "morphism". In other word, it is relationships between the objects that are really important in mathematics.)
A few points that I, as a non-mathematician, would like to elaborate on.
(1) Any set of definitions ends up relying on some smaller set of undefined things.
(2) There may be a very general and abstract definition of "mathematics" that is meaningful to mathematicians, but it will not be meaningful to students who have not studied any mathematical topics except the simplest.
(3) Definitions (and the related device of typologies) frequently have fuzzy boundaries but nice clear centers. So there is no ambiguity in saying arithmetic, algebra, geometry, and calculus are part of mathematics. I would be hard pressed to define "language" in a rigorous way that would command virtually universal assent, but the absence of such a definition does not undercut the utility of teaching various languages to students. I do not need a general definition of language in order to justify the teaching of French or Japanese or Mandarin or Hindi.
(4) It was said earlier that mathematics is broader than the study of numbers, which is true. But the study of numbers is one field within mathematics. The branches of mathematics that are mandated subjects of study involve numbers, which I consider incredibly useful mental constructs, and idealizations of the physical space in which we exist. Do you really believe it a serious question to suggest that there is no value in teaching people about these things because there may be no generally comprehensible definition that includes all the different fields comprising mathematics, including set theory, group theory, analysis, probability theory, number theory, etc., in addition to arithmetic, Euclidean geometry, elementary algebra, and calculus?
What about being really reductive. Math is the activity of creating a set of symbols, the rules for manipulating them, and then doing so. The particular application/inspiration and result is the product of inspiration and desire or curiosity.
Essentially I think of math as a game, including the game of making games. Math is valuable for several reasons, a) some math games parallel elements of the evolving natural world and provide information about future events or how to contrive future events, b) playing math games broadens the engaging mind's acuity, c) provides a means of communicating subtle and complicated ideas and observations, d) is fun for the oddballs. There are probably other benefits as well.
Regarding dog's and definitions. It seems to me that all definitions are agreements whose domain is always contingent upon the desired scope of the definition. For example, as I understand, a triangle was originally defined as a plane figure, when it is laid out on a sphere does it cease to be a triangle? Does one call this new figure a triangle or invent a new name? It seems the choice is arbitrary if there are convincing arguments for doing either.
A dog is dog if it meets the definition agreed by those who are party to the agreement. If later you learn about wolves, and hyena's, well, if your definition is loose enough, they are dogs, but, if upon closer inspection it is worthwhile to refine your definition, then perhaps they are not dogs but rather wolves and hyena's.
In short, it seems to me, a definition in a contrivance of meaningful communication and totally contingent upon present and possibly renegotiated agreements. What IS, is, but what we perceive and define is really just our best projection about the great IS, be our projections ever so humble.
Regarding math as contingent on pattern and symmetry. It does seem that after noting that there is a game to play the single fact that makes anything possible is, pattern, to use one word. If no pattern, then chaos, if chaos no math, right? (Isn’t chaos theory really the theory of “apparent chaos” ?)
Hmmm, no math if chaos … in the mind, or in the world? In the mind! An orderly mind in a chaotic world could still invent a game, math could exist even if it had no material benefit.
Math is a symbol game with benefits. My best estimation.