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This is pure Platonism. I reject Platonism in modern mathematics. Mathematics should, first, be used to describe the real world. (That is what scientists like

by javert 13y ago

This is pure Platonism. I reject Platonism in modern mathematics. Mathematics should, first, be used to describe the real world. (That is what scientists like Newton were using it for.) To do that, you don't need an axioms. Nor do you need beliefs. You only need observation of reality. It starts with the concept of a "unit," which can be added or subtracted from other "units." There is, of course, some prerequisite philosophy. ("Reality, you say?" "Why, yes, actually..."). Secondly, mathematics can deal with axioms and "games" and "universes" (to take words from the StackExchange post). But all of mathematics should not be subordinated to such a framework. I realize my argument is not "obvious." Fully establishing it would take a lot of work, and even then it would be widely rejected by the status quo Platonic mathematics establishment. I'm not claiming to do that in this comment. I'm just putting this alternative perspective out there for anyone who is interested.

adsr 13y ago
I don't see that at all, in fact the author in one of the comments admits to be a formalist, which is what I gathered from the answer as well.
EliRivers 13y ago
Mathematics should, first, be used to describe the real world. If by "real world" you mean the things and events we can directly witness, that seems to be deliberately crippling mathematics from the start and limiting it to things we already know about.
- javert 13y ago
  Depends on what you mean by "directly witness," but that's neither here nor there. What I am proposing would not cripple mathematics, it would simply replace "axioms" with _facts_ (such as, for a simple one, that adding one unit to another results in having two units). You can then get arithematic, algebra, calculus, and on and on. If people want to go off and define axioms for made-up worlds that do not necessarily correspond to reality and then go do math with them, that's fine. In fact, I would encourage it. But we should not be approaching _all of mathematics_ in that way, and that is my point. Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc.
  - gosub 13y ago
    > it would simply replace "axioms" with _facts_ You are describing physics.
  - fhars 13y ago
    "_facts_ (such as, for a simple one, that adding one unit to another results in having two units)" That is not a fact, that is just a consequence of certain sets of axioms, but not of others. It is patently wrong for addition in GF2, where adding one unit to another results in having zero units.
    - gjm11 13y ago
      Nope, not wrong in GF(2) at all. In GF(2), adding one unit to another gives you two units. It just happens that in GF(2) having two units is the same thing as having zero units.
  - jakobe 13y ago
    What's the difference between your "fact" and an "axiom"? It seems to be the same thing: a starting point from which you derive everything else.
  - EliRivers 13y ago
    Sounds like you want to take a subset of axioms that you like, label them "facts", and go from there. Presumably, you'll end up with a subset of mathematics. Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc. Can you show me what I get when I subtract three apples from two apples? How about showing me however many apples I need to square in order to get negative one apple?
  - gjulianm 13y ago
    The ZFC axiom set [1] doesn't define a made-up world. Actually, once you understand the axioms (they're somewhat difficult in notation and definitions, but the ideas behind are understandable) they make sense. There are some axioms made to avoid paradoxes (notably ZFC3 [2]), but the majority describe how we understand the world. It'd be difficult to imagine a world where two sets aren't equal even when they have the same elements, or where you couldn't construct a set that contains all elements of two other sets. How's this different to a set of facts as you say, like one unit + one unit is two units? It's not. I'm not sure of this, but I'd say you could construct a set of axioms based on those facts that is compatible with the ZFC axiom set. So, why choose ZFC? Because it's simple to define (how'd you formally define what is a unit, and what does the + operation mean?) and because it's more or less consistent with our world. We can extrapolate the things we find while doing mathematics to the real world. And, sometimes, we can discover things of the real world based on findings on things that don't exist. Complex numbers, for example: they don't exist, but they're really useful to define and explain electrical experiments [3]. We approach all of mathematics like this because its easier to study everything within the same system. And it's important that we keep a consistent underlying system so we can be sure that everything we discover based on the set of axioms and rules we've defined is true in a mathematical sense; and then those discoveries could describe our world. Wouldn't we have a consistent system, we could end up with contradictions. If we extracted mathematical concepts from the real world, we wouldn't be sure if a discovery we make from those concepts can be applied to the real world. TL;DR: Our set of axioms describes properly the real world, is simple and consistent enough and that enables us to infer new ideas that can be applicable to the real world too. [1] http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_th... [2] http://en.wikipedia.org/wiki/Axiom_schema_of_specification http://en.wikipedia.org/wiki/Axiom_schema_of_specification [3] http://en.wikipedia.org/wiki/Complex_number#Electromagnetism_and_electrical_engineering http://en.wikipedia.org/wiki/Complex_number#Electromagnetism...
saraid216 13y ago
> That is what scientists like Newton were using it for. Sounds to me like you're saying that mathematics is the Javascript to science's HTML.

by javert 13y ago

adsr 13y ago

I don't see that at all, in fact the author in one of the comments admits to be a formalist, which is what I gathered from the answer as well.

EliRivers 13y ago

Mathematics should, first, be used to describe the real world. If by "real world" you mean the things and events we can directly witness, that seems to be deliberately crippling mathematics from the start and limiting it to things we already know about.

javert 13y ago
Depends on what you mean by "directly witness," but that's neither here nor there. What I am proposing would not cripple mathematics, it would simply replace "axioms" with _facts_ (such as, for a simple one, that adding one unit to another results in having two units). You can then get arithematic, algebra, calculus, and on and on. If people want to go off and define axioms for made-up worlds that do not necessarily correspond to reality and then go do math with them, that's fine. In fact, I would encourage it. But we should not be approaching _all of mathematics_ in that way, and that is my point. Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc.
- gosub 13y ago
  > it would simply replace "axioms" with _facts_ You are describing physics.
- fhars 13y ago
  "_facts_ (such as, for a simple one, that adding one unit to another results in having two units)" That is not a fact, that is just a consequence of certain sets of axioms, but not of others. It is patently wrong for addition in GF2, where adding one unit to another results in having zero units.
  - gjm11 13y ago
    Nope, not wrong in GF(2) at all. In GF(2), adding one unit to another gives you two units. It just happens that in GF(2) having two units is the same thing as having zero units.
- jakobe 13y ago
  What's the difference between your "fact" and an "axiom"? It seems to be the same thing: a starting point from which you derive everything else.
- EliRivers 13y ago
  Sounds like you want to take a subset of axioms that you like, label them "facts", and go from there. Presumably, you'll end up with a subset of mathematics. Math has the same basis in perceptual concretes (i.e. "real things") as physics, biology, chemistry, etc. Can you show me what I get when I subtract three apples from two apples? How about showing me however many apples I need to square in order to get negative one apple?
- gjulianm 13y ago
  The ZFC axiom set [1] doesn't define a made-up world. Actually, once you understand the axioms (they're somewhat difficult in notation and definitions, but the ideas behind are understandable) they make sense. There are some axioms made to avoid paradoxes (notably ZFC3 [2]), but the majority describe how we understand the world. It'd be difficult to imagine a world where two sets aren't equal even when they have the same elements, or where you couldn't construct a set that contains all elements of two other sets. How's this different to a set of facts as you say, like one unit + one unit is two units? It's not. I'm not sure of this, but I'd say you could construct a set of axioms based on those facts that is compatible with the ZFC axiom set. So, why choose ZFC? Because it's simple to define (how'd you formally define what is a unit, and what does the + operation mean?) and because it's more or less consistent with our world. We can extrapolate the things we find while doing mathematics to the real world. And, sometimes, we can discover things of the real world based on findings on things that don't exist. Complex numbers, for example: they don't exist, but they're really useful to define and explain electrical experiments [3]. We approach all of mathematics like this because its easier to study everything within the same system. And it's important that we keep a consistent underlying system so we can be sure that everything we discover based on the set of axioms and rules we've defined is true in a mathematical sense; and then those discoveries could describe our world. Wouldn't we have a consistent system, we could end up with contradictions. If we extracted mathematical concepts from the real world, we wouldn't be sure if a discovery we make from those concepts can be applied to the real world. TL;DR: Our set of axioms describes properly the real world, is simple and consistent enough and that enables us to infer new ideas that can be applicable to the real world too. [1] http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_th... [2] http://en.wikipedia.org/wiki/Axiom_schema_of_specification http://en.wikipedia.org/wiki/Axiom_schema_of_specification [3] http://en.wikipedia.org/wiki/Complex_number#Electromagnetism_and_electrical_engineering http://en.wikipedia.org/wiki/Complex_number#Electromagnetism...

saraid216 13y ago

> That is what scientists like Newton were using it for. Sounds to me like you're saying that mathematics is the Javascript to science's HTML.