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Students: Let's come up with some crazy proofs based on our individual levels of understanding. Teachers: Let's do it by the book and come up (somehow) with co

Students: Let's come up with some crazy proofs based on our individual levels of understanding. Teachers: Let's do it by the book and come up (somehow) with conflicting answers. Mathematicians: Yeah, sorry guys. We made it all up. Pretty much captures most mathematicians I know.

Bahamut 13y ago
Definitions in mathematics are used to make the language describing abstract concepts elegant, as explained in the entry. However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.
- kazagistar 13y ago
  Sort of like how we use terse symbols for everything because ink and papyrus is expensive.
  - na85 13y ago
    And because if you're Isaac Newton and you're writing pages and pages of proofs, you naturally want to use a shorthand in order to save yourself time, energy, and from repetitive strain injury.
  - marcosdumay 13y ago
    Not really. We have some very good reasons for using the definitions we use every day. Those reasons are more related to us than our technology, although our tech is getting so good that we already need to think about it. And by the way, people changed almost all the usual definitions at the XX century. Now, mathematicians do throw the usual definitions away all the time. It's important to know when to reuse other people's coding, or roll your own.
- masterzora 13y ago
  My favourite algebra professor always said something to the effect of "we always have to be careful about abusing notation but if we make good choices of notation we can abuse it right." Definitions are sort of the same deal.
impendia 13y ago
Well of course we made it up. No one handed us any stone tablets with 0^0 on them. Mathematicians are always making definitions, and working out which of them should be kept and which should be discarded. We keep the definitions that make the most sense, that make our lives the easiest, that make theorems easy to state, that give math a sense of being natural. Indeed, in the early days of algebraic geometry there were big debates over which definitions to adopt. It is like deciding on a convention when you design a new programming language. In this case, experience has shown that it is pretty much always better to say that 0^0 is 1 and not 0. Among other reasons, there is exactly one map from the empty set to the empty set. But if you say 0^0 = 0, you don't get math blowing up in some big contradiction. You just get a little more kludge here and there, a few extra special cases of lemmas that have to be spelled out in more detail. Nothing too awful.
baddox 13y ago
Correct me if I'm wrong, but I infer that you're using "we made it all up" as a pejorative toward mathematicians. Of course mathematicians invented the terminology, notation, and methodology, but that's not a bad thing. It's a great thing, just like it's great that engineers "make up" bridges, chemists "make up" pharmaceuticals, writers "make up" novels, etc.
- dvanduzer 13y ago
  Mathematicians are engaged in a dramatically different kind of project than almost any other human discipline. Mathematical details emerge from definitions, but they appear exactly the same way for everyone else using the same definitions. And what's really surprising is how robustly those purely rational results compare to messy empirical reality. There is no good reason to believe that this should be the case! In most other human endeavors, when something doesn't work we just work until find something else that does work, and marvel at our ingenuity. Mathematics doesn't quite give you that option. There are right and wrong answers to questions that we create ourselves, but those answers are fixed once we ask the questions, even if we didn't know what they were. One enduring mystery in this vein was whether or not Euclid's fifth axiom was actually a logical result of the first four: https://en.wikipedia.org/wiki/Parallel_postulate#History https://en.wikipedia.org/wiki/Parallel_postulate#History
  - baddox 13y ago
    I don't think the fifth axiom is considered particularly mysterious anymore. The traditional fifth axiom clearly isn't a logical result of the first four, since it can be replaced with other parallel postulates to yield non-Euclidean geometries which are themselves perfectly workable and consistent. In fact, that section of the Wikipedia article notes that Beltrami proved the independence of the parallel postulate.
    - dvanduzer 13y ago
      > clearly It's mysterious because it is only clear in retrospect. The fact that you can negate the parallel postulate and get a system that is still self-consistent is incredibly mysterious. This is what sets mathematics apart. We "make it up" like other human accomplishments, and yet we can't actually just make it up. If we arbitrarily defined 0^0 = π, we'd just be speaking nonsense.
scythe 13y ago
0^0, like any indeterminate form, can be made to equal anything via sufficient cleverness. Consider the limit: y = lim_[x->0] x^[a / log(x)] We have log y = lim_[x->0] (a / log(x)) log(x) = a. So 0^0 equals any number at all! Of course, nobody in their right mind would define exponents this way, but the indeterminacy is inherent in the definition of the symbols.
- gohrt 13y ago
  How does your definition align with x^y in the case x!=0 or y!=0 ?

Bahamut 13y ago

Definitions in mathematics are used to make the language describing abstract concepts elegant, as explained in the entry. However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.

kazagistar 13y ago
Sort of like how we use terse symbols for everything because ink and papyrus is expensive.
- na85 13y ago
  And because if you're Isaac Newton and you're writing pages and pages of proofs, you naturally want to use a shorthand in order to save yourself time, energy, and from repetitive strain injury.
- marcosdumay 13y ago
  Not really. We have some very good reasons for using the definitions we use every day. Those reasons are more related to us than our technology, although our tech is getting so good that we already need to think about it. And by the way, people changed almost all the usual definitions at the XX century. Now, mathematicians do throw the usual definitions away all the time. It's important to know when to reuse other people's coding, or roll your own.
masterzora 13y ago
My favourite algebra professor always said something to the effect of "we always have to be careful about abusing notation but if we make good choices of notation we can abuse it right." Definitions are sort of the same deal.

impendia 13y ago

Well of course we made it up. No one handed us any stone tablets with 0^0 on them. Mathematicians are always making definitions, and working out which of them should be kept and which should be discarded. We keep the definitions that make the most sense, that make our lives the easiest, that make theorems easy to state, that give math a sense of being natural. Indeed, in the early days of algebraic geometry there were big debates over which definitions to adopt. It is like deciding on a convention when you design a new programming language. In this case, experience has shown that it is pretty much always better to say that 0^0 is 1 and not 0. Among other reasons, there is exactly one map from the empty set to the empty set. But if you say 0^0 = 0, you don't get math blowing up in some big contradiction. You just get a little more kludge here and there, a few extra special cases of lemmas that have to be spelled out in more detail. Nothing too awful.

baddox 13y ago

Correct me if I'm wrong, but I infer that you're using "we made it all up" as a pejorative toward mathematicians. Of course mathematicians invented the terminology, notation, and methodology, but that's not a bad thing. It's a great thing, just like it's great that engineers "make up" bridges, chemists "make up" pharmaceuticals, writers "make up" novels, etc.

dvanduzer 13y ago
Mathematicians are engaged in a dramatically different kind of project than almost any other human discipline. Mathematical details emerge from definitions, but they appear exactly the same way for everyone else using the same definitions. And what's really surprising is how robustly those purely rational results compare to messy empirical reality. There is no good reason to believe that this should be the case! In most other human endeavors, when something doesn't work we just work until find something else that does work, and marvel at our ingenuity. Mathematics doesn't quite give you that option. There are right and wrong answers to questions that we create ourselves, but those answers are fixed once we ask the questions, even if we didn't know what they were. One enduring mystery in this vein was whether or not Euclid's fifth axiom was actually a logical result of the first four: https://en.wikipedia.org/wiki/Parallel_postulate#History https://en.wikipedia.org/wiki/Parallel_postulate#History
- baddox 13y ago
  I don't think the fifth axiom is considered particularly mysterious anymore. The traditional fifth axiom clearly isn't a logical result of the first four, since it can be replaced with other parallel postulates to yield non-Euclidean geometries which are themselves perfectly workable and consistent. In fact, that section of the Wikipedia article notes that Beltrami proved the independence of the parallel postulate.
  - dvanduzer 13y ago
    > clearly It's mysterious because it is only clear in retrospect. The fact that you can negate the parallel postulate and get a system that is still self-consistent is incredibly mysterious. This is what sets mathematics apart. We "make it up" like other human accomplishments, and yet we can't actually just make it up. If we arbitrarily defined 0^0 = π, we'd just be speaking nonsense.

scythe 13y ago

0^0, like any indeterminate form, can be made to equal anything via sufficient cleverness. Consider the limit: y = lim_[x->0] x^[a / log(x)] We have log y = lim_[x->0] (a / log(x)) log(x) = a. So 0^0 equals any number at all! Of course, nobody in their right mind would define exponents this way, but the indeterminacy is inherent in the definition of the symbols.

gohrt 13y ago
How does your definition align with x^y in the case x!=0 or y!=0 ?