Does Mathematics FASCINATE You?

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Is Mathematics Beautiful?

1. Bertrand Russell (1872-1970), Autobiography, George Allen and Unwin Ltd, 1967, v1, p158
It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect but true.

2. Bertrand Russell (1872-1970), The Study of Mathematics
Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

3. Aristotle (384 B.C.-322 B.C.), Poetics
Beauty depends on size as well as symmetry.

4. J.H.Poincare (1854-1912), (cited in H.E.Huntley, The Divine Proportion, Dover, 1970)
The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

5. J.Bronowski, Science and Human Values, Pelican, 1964.
Mathematics in this sense is a form of poetry, which has the same relation to the prose of practical mathematics as poetry has to prose in any other language. The element of poetry, the delight of exploring the medium for its own sake, is an essential ingredient in the creative process.

6. J.W.N.Sullivan (1886-1937), Aspects of Science, 1925.
Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of Mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.

7. G. H. Hardy (1877 - 1947), A Mathematician's Apology, Cambridge University Press, 1994.
The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

8. Lawrence University catalog, Cited in Essays in Humanistic Mathematics, Alvin White, ed, MAA, 1993
Born of man's primitive urge to seek order in his world, mathematics is an ever-evolving language for the study of structure and pattern. Grounded in and renewed by physical reality, mathematics rises through sheer intellectual curiosity to levels of abstraction and generality where unexpected, beautiful, and often extremely useful connections and patterns emerge. Mathematics is the natural home of both abstract thought and the laws of nature. It is at once pure logic and creative art.

9. I.Newton, Letter to H.Oldenburg, the Secretary of the Royal Society, October 24, 1676, in A Source Book in Mathematics, D. J. Struik, ed, Princeton University Press, 1990
I can hardly tell with what pleasure I have read the letters of those very distinguished men Leibniz and Tschirnhaus. Leibniz's method for obtaining convergent series is certainly very elegant...

10. Jane Muir, Of Men & Numbers, Dover, 1996.
Gauss: You have no idea how much poetry there is in the calculation of a table of logarithms!

11. F.Dyson, in Nature, March 10, 1956
Characteristic of Weyl was an aesthetic sense which dominated his thinking on all subjects. He once said to me, half-joking, "My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful." (Herman Weyl (1885-1955))

12. O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956
To Goethe again we owe the profound saying: "the mathematician is only complete in so far as he feels within himself the beauty of the true."

13. O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956
"A mathematician," said old Weierstrass, "who is not at the same time a bit of a poet will never be a full mathematician."

14. Jakob Bernoulli, Tractatus de Seriebus Infinitis, 1689 (quoted in From Five Fingers to Infinity, F.J.Swetz (ed), Open Court, 1996)
So the soul of immensity dwells in minutia.
And in narrowest limits no limits inhere.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!

Math, much like any subject, has a pretty side and an ugly side. Most people don't see the pretty side and I'm lucky enough to have seen it. I love math! (I also like the "ugly side".)

Not me.

I disbelieve that Mathematics has an ugly side. I also disbelieve that a majority of people have been exposed to math.

What is taught in standard schooling is not math, math is the field of understanding, of logic, of finding truth, and not of memorizing that 1+1=2. Math is a field that a computer can't do (though can help in (see the four colour theorem)).


(I think the fact that I'm applying to PhD programs in math answers the topic question )

Math can definitely be beautiful. Here are some images I created about 10 years ago using a fractal program.

Constructing and calculating Taylor and Fourier series is plenty ugly, believe me.

Don't even get me started on solving n-th order non-linear differential equations....

Taylor and Fourier series aren't ugly. I'm not a fan of working with them myself (seeing as I focus in discrete math, that's not surprising), but that doesn't make them ugly. The fact that I can't stand working in statistics doesn't make it ugly either. (And yes, I've had to deal with them before).

I love math. When I say an equation or a proof is "beautiful", I mean it.

This, especially:

e^(i*pi) + 1 = 0

If you don't get it, try assuming that the exponential is polar coordinates for a complex number...

Beautiful, right?

I'm not a mathematician and probably never will be. It's just a hobby. But I love the pure logic of it.

Oddly enough, I sucked at arithmetic to the point that I didn't know the multiplication table until high school.

That is pretty cool. Have you seen this?

e^(i*pi) = -1

and so

e^(i*pi) = i^2

Kind of weird! Do you know what it *means*? It's so elegant and yet so odd how mixing two irrational numbers with a self-defined imaginary number would result in this weird equality that must geometrically mean something.

I really into the theoretical foundation stuff. The topic in mathematics that fascinates me the most is iteration and recursion.

I used to love it, but I'm going off it a bit now. I don't think there's any "truth" to maths, it is a set of abstract fictions created to serve society.

Here is a theory of mathematics that I wrote and posted on my tumblr:

"Theory of Mathematics
Rant time! This one isn’t as trollish as the others, and it not going to be as much of a textwall, but hopefully no one will understand it and they will just get intimidated and skip to the TL;DR where I will troll them. I guess this is a little like the three assumptions in Pi.

1. Math is made of thingies called objects, which can be pretty much anything. The numbers 80085, e and four, the set {0,e}, the function f, the variables x and q, the ordering(?) (1, 2, 3, 4), the string “Your mom, lol”, the character J, the category matrices (as well as the category categories), the identity Sexdx=ex, the equation x = 1, the axiom/definition ”A number equals itself” and the vector <1,2> are just a few objects. A picture of a cow could probably defined so that it is an object, which leads to the second point.

2. Mathematical objects are created by defining their basic properties, and as long as these properties do not contradict each other objects can be defined to have pretty much any properties. I guess properties themselves are mathematical objects as well. ex. i is defined as a number so that i2 = -1, but it cannot be defined as a real number because the way real numbers are defined prevents any real number from having i’s defining property. An extension of the addition operation is then defined so that it retains the same properties on i as it does on the real numbers. Also I can define the “Your Mom” object, the “Me” object, and the “+” operator so that “Me + Your Mom = Fun”.

3. More properties are discovered in objects based on their basic properties. ex. from i’s basic properties every we know about complex numbers where discovered

4. Mathematical objects can be used to model physical things and more more abstract ideas, including other mathematical objects. ex. Quantities can be modeled with numbers (ex. temperature in degrees Celsius), the concept of number can be modeled, by, well numbers (duh), electrons can be modeled by functions called wave functions (according to current scientific theory, which may just be BS), colors can be modeled by complex numbers and a shade number, the picture of said cow may be modeled by a complex function, and the number ten may modeled by the string “10” when using the system (I’m assuming systems are mathematical objects too) called Decimal

5. An object called a theory is based on a few objects called postulates, and these postulates are used in conjunction with definitions to discover objects called “theorems”. Theories are used to model a lot of different stuff that may actually have meaning to us. Some theories are based upon other theories. Ex. Euclidean Geometry is a theory based on Euclid’s postulates. Its used to model lines and crap in our world. Arithmetic is also a theory, but I still trying to decide exactly what postulates its based on. Most theories used are based on Arithmetic and Logic.

TL;DR: Math is a bunch of bull crap made up by a bunch of old white guys. "

Its very crude and a bit rude, but that's the nature of my tumblr, its meant to be trollish and silly and philosophical and random at the same time. I think I should revise it to say something about the system/theory of logic. So yes, Math is a set of of abstract fictions created to serve society, but they are created so that as long as they remain abstract they are true, and how true it is for concrete objects depends on how well the mathematical model you are using models reality. According to Plato Math is not a perfect representation of our world, and that's true, at least for most mathematical models as they over simplify many things.

I wish I knew enough to understand that...

Non-linear differential equations are nasty in general. Only a small number can you even solve analytically. If you're a scientist you hardly ever solve differential equations by hand. You program a computer to give an approximate solution. Of course, even numerical approximations can be nasty when the equations are such that numerical errors blow up on you.

Taylor and Fourier series aren't that nasty though. The fast-fourier transform is actually quite amazing.

I think of complex numbers as two dimensional vectors with a neat multiplication property that works out such that the angles relative to the positive real axis add together while the lengths are multiplied.

I do like math but Im not always the best at it. I like numbers and puzzles, its like I need to figure this problem out. Im good at high school level math but not calc and beyond. Even if Im not always the best at upper level math, I still enjoy it even though I make a lot of mistakes.