Mathematics: created or discovered?

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Creativity is not an emotional feeling. Mathematicians need to figure out what to do with what they already know. If what they're doing isn't straightforward or obvious, they need to be creative to figure that out.

Maths and science, to me, is just a different interpretation of the world. The truths discovered in both disciplines are expressed in a particular language.


This is exactly the kind of thing you do not learn in school. There you just learn about applied Maths, nothing about Maths itself. This is why I could never accept Maths as a subject I enjoy.

I don't approve of this creativity/logic binary because both can complement each other.

It's generally accepted that mathematics is the language of science. That is, if a scientific principle can not be expressed in mathematical terms, then it is not likely to be science - generally speaking.

Eh, for some branches of science. It's hard to express biology in mathematical terms, for example.

Anything with some solid proof (or in some cases solid logic) behind it can be considered scientific.

An interesting discussion.

I am fascinated by the Phi, Golden Mean Ratio. Not only is it unique, so much in nature and science seems to fall into place around it. I'd need a good explanation to show me how it was not just discovered.

I don't find it so unlikely. What I'm saying is that the common objects of study in mathematics are common objects of study precisely because they make pictures in your head when you're studying them. The reason for this is that they describe things that we can imagine. Things that we can imagine are the sort of things that we have experienced, i.e. the concepts which "fit" the universe. So mathematics which is built on these objects is just the sort of mathematics which you might "expect" to have an unexpected application.

I would be very surprised if someone randomly generated a list of axioms, found a bunch of consequences, and the consequences turned out to be useful for the real world.

I don't understand you Declension. Modern Physics is often something which can often only be understood in mathematical terms. Visualization may well out of our cognizance. I think that certain things, like the constants in maths are givens. Did Fibonacci discover or invent those numbers? However, many theories may only be approximate models skirting around the edges of truth and since there are many times disagreement between the great experts, often political, who is to say if they are true.

Good question.

I would say discovered because as people have already pointed out that a lot of maths requires creative thinking, at the end of the day the maths has to add up correctly at the end of the creative process.

So the creativity on display from mathematicians is different to the creativity on display in an artistic sense because an artist can create a painting or sculpture and the quality is subjective once the creative process is finished. For the mathematician the end result of the creative process can be measured absolutely.

Both. Mathematics was both created and discovered. I'll explain that in depth later.

Once a structure is formulated (an act of creation) finding its logical consequences is an act of discovery. So the answer is --- both. Created and Discovered.

ruveyn

Yes. Also, mathematics can used to model anything, both art and science, and while the patterns are discovered the initial systems are created.

For those who say creativity is not for math, it's for art, think about this. A piece of art was created digitally. That means it can be stored digitally. It can therefor be stored in a sequence of on and off bits, which can be represented by a binary and therefor natural number. Now, is the artist creating a work, or are they just discovering a number?

I'm actually leaning more so towards created than discovered. The way how every mathematical discipline works is that it takes a few axioms, and discovers the consequences of those axioms if there were true. In geometry, we were given postulates, and never really proved them because they were consider so rudimentary and obvious (and indeed, the main difference between theorems and postulations were effectively the former were proven while the latter were taken for granted). But axioms are not proven, they are taken for granted, and often all mathematical proofs are made with the contingency 'so long as this axiom applies.' So in a sense, the most rudimentary basis of math is created by mathematicians to create a basis on which to discover theorems, and that is nevertheless a basis created, not discovered.

But, that could be iffy, since there are very intuitive reasons why we choose our axioms. Such as closure under addition simply comes from the fact that, in the real world, if you have an apple and group it with another apple, you now have two apples. But this sort of 1+1=2 is not always true, such as in the case of computer binary systems, where 1+1=10 in binary, IIRC. Still, it's through this observation we find other sub-disciplines, such as fractal geometry which was taken seriously after Mandelbrot drew connections between his notions of fractal geometry and nature. Even so, I feel it's worth bringing up that the most rudimentary premises of mathematics aren't necessarily proven as much as taken for granted, and thus one could argue that those axioms that we use to discover everything else in mathematics were not discovered, but rather created (or perhaps observed). I'll just leave it at that before I make this even more complicated, especially since it seems to be semantics more than anything.

I don't think all of mathematics (or physics) is easy to mentally picture. How do you picture an infinite dimensional vector space? Can you usefully picture a 17 dimensional cube? When you think of an electron orbiting an atom, you might picture a little orange ball orbiting a clump of bigger balls, which is in accordance with our everyday experience but not with physics.

What you said does apply to some math that was invented/discovered by people trying to do physics. I don't believe it applies to all math.

I do it by picturing a three-dimensional vector space and a three-dimensional cube, and learning from experience the situations in which this picture is misleading. They make pictures in people's heads. They're the wrong pictures, but they are pictures.

I think that all mathematics that people study is either (1.) ultimately an extension, or built on, something that they can easily imagine, or (2.) created after it was discovered that it was necessary for science. I mean, what other reasons would there be for studying something?

I guess a more accurate picture of electron orbit is as a "cloud" of possible geometric configurations of the electron relative to the other particles in it's immediate vicinity, the cloud being the most dense where a particular electron is most likely to "collide" with or trade energy with something else.

This makes about as much sense as saying "Biology, created or discovered".