Remember--This could happen if he did it wrong: http://www.davidairey.com/focus-on-art-dan-havel-dean-ruck/
hey everyone, i have an assignment for class [ToK], you can all help me! please answer these questions on a reply, and if you can include your degree:
1 Is Math invented or discovered? [examples]
2 Where is Math? [everywhere IS an acceptable answer, but consider feelings http://imgs.xkcd.com/comics/useless.jpg ]
3 [if possible] how is math knowledge generated?
4 How is math useful?
you opinion is super important to this assignment, PLEASE :D!
I do not see why I should be doing your assignment for you, but this is interesting enough I want to put my 2 cents in.
1) Maths exhibits invention discovery duality. Frameworks, notation, axioms and postulates need to be invented, while plenty of discoveries can be done within. Examples, from largely my field of physics, include the invention of proof systems (Euclid), algebra, calculus, idea of relativity and so on. Within it, discoveries like the irrationality of Sqrt(2), generalisability and flexibility of mathematics, usefulness of infinitesimals and infinity, E=mc^2 and so on are made. PA would be a great framework that I would call invented, although it has strong similarity to previous frameworks that it arguably evolved from. Within it, Cantor's invention of the proof that the cardinality of the rational numbers is the same as the natural numbers led to the startling discovery of the largeness of the transcendental numbers (so huge the cardinality changes.
2) Maths is wherever operations are (here, I mean the idea of operational definitions vs philosophical/dictionary/encyclopedic/layman/whatever else that does not require maths definitions, as the idea was proposed by Newton). As long as we stay within logic, rationality (not the numbers), computation, measurement, maths is pretty much anywhere in this restricted space. (I think this is a long enough answer for some serious number of marks, unless you need to write an essay on this.)
3) I only know of 2 methods where the boundary of mathematical knowledge gets pushed back, and it seems inconceivable to have any others. They are intuition and formal proving, of which neither is the one true answer -- intuition goes too fast and tends to go wrong, and maths is not well equipped to deal with inconsistency. Formal proving is too slow to be of much use, although Godel's incompleteness theorem was a huge win for formal proving. Historically, the mathematical climate swings between the two, and it seems clear that a merger of the 2 is required really. In physics, at least we can take data and fit curves to them to get a preliminary answer. In maths, there has not been much of that happening after Euler. Not plausible that it would be of much use in the future either.
4) How is maths useful? You do understand that it is becoming increasingly clear that the progress of humanity itself is rather strongly dependant on the progress of mathematics itself, do you not? It is like, we physicists keep running into dead ends we need help with the maths/notation, and we either invent quick and dirty notation for mathematicians to clean up, or we whack them into working harder. Or else, humanity just slows down progress until we find some breakthrough again.
Not to mention that a huge part of maths is logic itself, and hence all rational thought needs maths.
Otherwise, applied maths almost rules the usefulness space, although that would be rather sad -- the other branches of maths are just as beautiful, intricate and important to life, just probably not daily life, that is.
Yeah, my 2 cents weigh more.
LOVE you answer, and the assignment is asking people, not answering the questions :P They will be compared with the answers of other people like literature teachers and so on, to see what the perception of math is from different areas.
Your answers were oh but so beautiful! i LOVED them, thank you very much!
1. Math is invented by humans to find and express the underlying order in the world (examples: counting, calculating, geometry, group theory, calculus, ad infinitum) but then it also leads to discoveries of insights on further relationships and forms of order in the world (example: e^πi=-1, or then the application of complex numbers to electromagnetic fields. Another example: the Egyptians knew the 3-4-5 triangle had a perfect right angle. The Greeks (Pythagorean theorem) realized that any triangle in which a^2 + b^2 = c^2 is a right triangle.)
2 Math is found or applicable anywhere that ideas can be given a mathematical value, that is, a more or less precise value that can then be used in mathematical operations. This would include quantities (money or votes), truth values (logic), topological invariants, and so on. Again, the list seems endless.
Feelings cannot be assigned such values, and so resist mathematical analysis. In this sense, human persons can understand and use math, but math cannot fully express or analyze the human experience.
3 Math knowledge is generated by people using intuition to create interesting and elegant systems of "mathematical" values--again, see number 2--and to discover or develop new insights and relationships. Ideally, logic is then applied to show that these insights actually are true, or to find where they fail (and why).
Having then advanced mathematical knowledge, the process is repeated, now with additional possible systems, ideas, relations, and analogies to throw into the mix.
4 How is math useful? It has proven remarkably useful in describing many aspects of the world; there are very few mathematical ideas, however esoteric and strange, that have not been applied to the world in some way. Once applied, it helps us to understand and predict more of the world than otherwise we would be able to.
The limitations of math are that we come to expect that all (or almost all) things are subject to mathematical analysis and logic, and are astonished when it proves otherwise.
1. Mathematics is the language of the universe - it is discovered. The exact techniques, syntax, notation are man-made: consider our decimal and binary systems of numbers. They use digits and weights. The Romans used different symbols for all the new scales of magnitude. Such systems are invented and are merely protocols used for conveying a abstract ideas. If you were asked to picture the number "2" you'd draw two dots, the symbol, or something. But it's impossible to picture the IDEA of 2. Or any number.
2. Maths is in the very fabric of the universe. Maths is, in a way, the language used by Physics, Chemistry, Biology, Computer Science, Engineering. It's the language of Science. So ask yourself "where is language" in the context of science.
3. Magic inspiration particles flowing through space and time like little neutrinos. Should they hit the wrong target, such as a duck, humanity loses its chance for creating something profound, and duckdom gains... a really confused duck.
4. The computer you used was based on quantum physics of the semiconducting transistors. Communication needed to share your request is based on Fourier's Signal Analysis.
The satellites floating above us have been launched using the inverse square law.
From control engineering to management, from communication to programming, from accounting to mechanics, from computer science, through demographics to politics, from cartography to mechanics and economics; all of the things that can't work without a solid foundation - all of them rely on maths. Because maths is a way of capturing reality.
1. Math is discovered. Math is a collection of truths. These truth are true whether or not we know them. Example: When no one knew about pi, what was the ratio between the diameter and circumference of a circle?
2 Any system with rules and regular behavior can be analyzed by math. So math if found in all non-random systems. But then we can use probability theory to study random systems, so it's present there as well. Everywhere :)
3 I don't think generated is the right word... math knowledge is obtained by proofs. Until you prove something, it's not math, it's just speculation.
4 Math is a language with which we can describe the world around us. Through it, we can discover hidden truth about the world, and use these truths to improve our quality of life. It is also useful for training us to think logically.
...The International Baccalaureate holds very strict views towards collusion and malpractice, you know.
the assignment was to ask people [teachers preferably] from different areas on their personal answers to these questions, for the results to be compared later.
it is an assignment on local level [as in regarding my high school] and not for IBO level, so i'm doing nothing illegal, but yes they're pretty strict.
1) The semantic distinction between "invent" and "discover" is null when applied to a Platonic universe of abstractions.
2) The fantastic progress of physics following its mathematization strongly hints that everything is math. Sometimes we can't solve the math. Sometimes we can't setup the math. Sometimes we don't even know how to represent the process so that we can express it mathematically. But we are advancing on all three problems.
3) On the data-information-knowledge-wisdom axis, you have asked how that portion of mathematics which has be contextualized beyond the level of trends, into the level of frameworks is generated. Typically this is accomplished by the concerted effort of many mathematicians. Euclid's Elements is at this level, as is the classification of finite simple groups, also the Langlands program.
4) It isn't. Q.v. the invention of the steam engine, the prediction of the positron, most of the 20th century, and a few other examples.
I'll start by noting that all of your respondents, so far, have taken an "applied math" viewpoint; since my own background is in "pure math" (generalized topological algebra). To me the responses seem to miss the point that the questions amount to "What is mathematics?". My answer is "The formal study of formal systems." Thus the discussion has focus on those items of interest to mathematics rather on mathematics itself. That said:
-- 1 Is Math invented or discovered? [examples]
No. neither term is really applicable.
-- 2 Where is Math? [everywhere IS an acceptable answer, but consider feelings]
Nowhere. Since math is an abstraction of abstractions, "place" has no meaning. The bracket suggests the intended question would have been "Where is Math applicable?" The answer to this would be indirectly everywhere since any area of study can be formalized.
-- 3 [if possible] how is math knowledge generated?
Someone noting similarities between two "unrelated" areas formalizes them using similar structures. The nature of those structures is thn abstractly studied, possibly by someone else.
-- 4 How is math useful?
In a strict sense it isn't. Mathematics is a fine art; any use made of its results is incidental.
2in the universal constants, ergo, everywhere.
4computation of mathematical principles governs our society's infrastructures and influences all others.
short and concise, nice; thanks a lot!
Just wanted to say I love this comic. I only get like, 1 in 10 of them, but they make me think and are fun!
is no one gonna make an Arthur Benjamin reference?
Ah, but can you square a five digit number in your head?
there... happy now?
Squaring a 5 digit number is easy, if I get to pick the number (I'll pick 10000).
I just knew someone was going to say that as soon as I posted my comment! Should have said "...a random, 5 digit..."
Yes: *one way* of writing the square of 47293 is 47293^2. There are of course other ways, but yes - I did this squaring in my head and hereby present this number in one of its representations.
Why are you all looking at me?
so ronery. ;_;
To highschoolnerd (I am an undergrad applied math/biology student):
1. Discovered. We may construct systems of axioms that seem "invented," but we don't just choose axioms arbitrarily. Our constructed formal systems are motivated by a desire to reflect the structure of abstract mathematical objects.
2. Everywhere, including feelings. Google 'Steve Strogatz Romeo and Juliet' for love described in differential equations. All things can be described in mathematical terms.
3. By intuitive leaps, which are checked and refined by formal proof.
4. In more ways than could possibly be listed. From models of cancer, ecosystems, and the spread of epidemics to the foundation for physics that makes all our modern technology possible, math finds application in virtually every human pursuit.
You said mathemagician?
Dear highschoolnerd, and others reading these comments,
I am a philosopher of science. Whilst the above answers are very interesting, they're of course a bit short. I'd like to point out a nice book and a nice article you could read if you're really into this: Introducing Philosophy of Mathematics, by Michele Friend, Acumen Publishing (2007), and What Numbers Could Not Be, by Paul Benacerraf, The Philosophical Review, 74:47-73 (1965). Both are rather serious and you might have to skip over half if you're still in high school, but I still think you can catch the drift.
But, as highschoolnerd pointed out, the assignment was to ask others how they would answer the questions, not for the students to come up with or find the answers.
thank you Rose, I'll try to get hold of those books for my spare time, they seem interesting and seem like something i'd read
Anyway, as bmonk stated, the assignment was for others to answer the questions [teachers at my school, actually, but i knew i'd get some cool responses here too :D] but thanks for the book titles, that topic seems interesting, ♥ you all!
I know, but I thought you might want to read a little more anyway :) Hope you like it!