What is a number?

[nihilistic]

you may as well ask why -1 * -1 is 1 not -1, an issue of mathematical consistency.

Well, there is actually a very simple proof of that identity.

So 0! or 0 factorial=1 for the sake of mathematical consistency.

A factorial is defined as all numbers above one that can be timesed by itself -1 at least by summation rules, but in general it equals (n + 1)! = n!(n + 1) which allows for 0.

this definition makes many identities valid for zero sizes. In particular, the number of combinations or permutations of an empty set is, clearly 1.


0!=1
1!=1*1=1
2!=2*1=2
3!=3*2*1=6

same thing seems to go for 0^0=1 or 0/0=? we can't have our cake and eat it it seems.

the power rules wouldn't work if ^0 did not =1 e.g. 0^0*2^2=4 or just 2^2. If 0=0 then it equals 0, there dead simple.

power rule (x^m)(x^n)=x^m+n

Now that couldn't be more wrong. 0! has nothing to do with 0/0 or 0^0. 0! is simply defined to be 1 in the factorial function. 0/0 and 0^0 are both the same thing, indeterminate.

Division is defined as a preimage function of multiplication, but since multiplication by 0 is not a one to one function, division inherits both underdetermination and overdetermination problems there. 5/0 is undefined because there is nothing which when multiplied by 5 equal 0. 0/0 is indeterminate because anything multiplied 0 is 0.

0^0 s indeterminate because it is essentially 0/0. And 0^k where k<0 is undefined, because it is essentially 1/0. Power rule or whatnot still holds as you can see.

 

[nihilistic]

Nobody's defined the numbers from Ø yet? I suppose I can do some of it from memory, and then newfangle or some other mathematician that hasn't forgotten half of their training will correct me.

First of all, there is no precise equivalent to a number in the world. While we might say "1 dog", there's some point where cells start being a dog and some other point where the dog ceases to be a dog and becomes rot.

So we start with the empty set, written Ø or {}. The number of elements in the empty set is 0. Then we notice that we have a set, and the number of sets we have so far is 1, or rather that the set {Ø} or {{}} contains 1 element.
Now that we've gotten underway, we can define the set {Ø, {Ø}}, which has 2 elements. This lets us define {Ø, {Ø}, {Ø, {Ø}}} which is a set with 3 elements. And so on.

Sure, it can be constructed that way. However, the set theoretic construction is not necessarily a "definition" in every aspect, i.e. the numbers are still interpreted as quantity and rarely as sets. What you have constructed, is a system involving sets and set operations that is isomorphic to the natural numbers ad its operations. Items that are isomorphic must share the same mechanics but can have different interpretations. Math though, is all about the mechanics.

 

[Sidhe]

Well, there is actually a very simple proof of that identity.



Now that couldn't be more wrong. 0! has nothing to do with 0/0 or 0^0. 0! is simply defined to be 1 in the factorial function. 0/0 and 0^0 are both the same thing, indeterminate.

Division is defined as a preimage function of multiplication, but since multiplication by 0 is not a one to one function, division inherits both underdetermination and overdetermination problems there. 5/0 is undefined because there is nothing which when multiplied by 5 equal 0. 0/0 is indeterminate because anything multiplied 0 is 0.

0^0 s indeterminate because it is essentially 0/0. And 0^k where k<0 is undefined, because it is essentially 1/0. Power rule or whatnot still holds as you can see.

I explained it better later on by explaining that it must equal 1 in binomial theory. But I take the point. I did admit that I'd done a terrible job in trying to explain why seemingly arbitrary rules existed and then quoted some people who could explain what I was saying better than me.

 

[Frasco]

I'll place my piece to this useless puzzle

A number as defined symbols (0-9) are just part of the language we use. However a number is much more than a graphic and has a meaning implied. Such meaning is mearly a magnitude and a magnitude is a count of something (or anything since it has no useful meaning unless paired with a unit).

The inner question, I believe is why Maths work? By which universal coincidence can mathemathics be aplied to model our universe? The matter is more philosophy than Mathemathics.

I would say it is part of how we have evolved as a civilization. Our mind works in a manner of numbers and these came out of our hands (10 fingers total, 10 numbers total).

We need, in every aspect of our understanding to compare everything to a base in order to comunicate. We compare feelings in every one of our senses to a base learned in our first years (colors, flavours, cold or hot, good and bad...).

The number came as something much more abstract and was necesary for communication and thus could model simple systems... 10 fingers is a system and when the number is defined anyone can imagine the system to a degree of exactitude when they compared to their own 10 fingers (the base).

Of course our curiosity made us go further, and it is said that the last purpose of physics (which was the next step after mathemathics had evolved enough) is to describe a perect model of the whole universe.

A little piece at least.

 

[warpus]

A number is best explained using language theory type thinking.

Numbers are words of the mathematical language that we use to form mathematical sentences.

I don't see a need to get more philisophical than that.

 

[Ayatollah So]

Ostensive definition: a number is one of these kinds of things:

Keep in mind that, first of all, we have the natural numbers: 1, 2, 3, 4, 5, 6, etc. (0 can be included too, depending on the definition of natural number we're using). But that, of course, is not all: we have to expand that to the integers, which include negative numbers. Then we expand that to the real numbers, which include irrational numbers. Then we move into complex numbers. There are also quaternions, octonions, and various other sets of numbers.

Cute how I used your question to answer itself, eh? Oh, and I only read page 1 so far, so if anyone beat me to it ... well then, rats!

If a kid never learned math this way, and instead just manipulated abstract symbols right from the start, how could these symbols have any meaning to him?

Cool question! I suspect that this kid would be less likely to become extremely good at math - the real-world connections probably provide some insight. I have no evidence whatsoever for this hunch.

 

[Ayatollah So]

Well, I'll add my two cents (whatever "two" is ).

In any axiomatic system, you have to have certain primitive concepts (also called undefined terms). Primitive concepts aren't defined, because otherwise you'd just keep defining concepts in terms of other concepts, ad infinitum.

I like this response a lot. I think it fits well with resorting to ostensive definition.

In other words: rats, Petek beat me to it.

 

[WillJ]

I think my question/thread is really concerned with two different things. One is providing a reasonably deep, perhaps formal, definition of "number." The other is understanding how these "numbers" fit in with the world.

I don't think any of the definitions of "number" given here so far have been very satisfying. For example, with regard to Erik's set theoretic defintion, I agree with nihilistic:

Sure, it can be constructed that way. However, the set theoretic construction is not necessarily a "definition" in every aspect, i.e. the numbers are still interpreted as quantity and rarely as sets.

So, since there doesn't seem to really be a good definition out there, I am going to have to agree with Petek:

In any axiomatic system, you have to have certain primitive concepts (also called undefined terms). Primitive concepts aren't defined, because otherwise you'd just keep defining concepts in terms of other concepts, ad infinitum.

I think this is the best possible answer. Like you said, Petek, in the world of mathematics, there must be undefined primitive concepts just as there must be unproven axioms, and I guess "number" is a perfect place to start. And, like you said, if there is actually a suitable definition of number, it'll inevitably require some other primitive concept; for example, Erik's definition leaves "set" as a primitive concept. So yay, you win.


...I think that settles problem #1, but there's still the other half of the mystery (which is perhaps a completely separate mystery, and has nothing to do with what a number is---or maybe it does?), which arises from the fact that mathematics is paradoxically both perfectly concrete and perfectly abstract. Its truths are a priori, yet they can elucidate reality. As Birdjaguar pointed out, mathematics rose out of the need to count things, and that is still the way that (almost?) everyone on this planet is first exposed to math. Some intellectuals even go so far as to dismiss the idea that mathematical truth is in its own little a priori world: in the words of the mathematician Vladimir Arnold, "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." (See here.) Could he be right?

One way to think about this is to look directly at the axioms of math. Are the axioms just arbitrary starting points? Or is there a good reason to use them---perhaps the axioms are actually true?

Most people probably don't think it's much of a mystery that the world can be described mathematically; they just take it as a given, as the natural order of things: obviously physics and math are connected; that's just how it is! But there are certainly people who marvel(ed) at the fact:

Albert Einstein: "How can it be that mathematics, being after all product of human thought which is independent of experience, is so admirably appropriate to the objects of realtiy?"

Richard Feynman: "Mathematics is not real, but it feels real. Where is this place?"

Now, if these two brilliant theoretical physicists think there's a mystery, there damn well is a mystery! For more on it, see here. Earlier, I brought up a thought experiment, which I don't think anyone commented on: Imagine we lived in a different universe. (Of course, if the universe were different, we would be different, but just ignore that.) Now, is it possible that, living in that universe, we might be unable to describe its properties mathematically? Or would we always be able to use math to describe the universe, no matter what it's like? Is "using math" simply a matter of finding patterns, and we humans could find patterns in anything---and so the connection between math and reality is, after all, unremarkable?

------------

Proof by induction is rigorous, and has plenty of uses. In fact, one way to define the set of natural numbers is by induction.

I'm assuming you're talking about this kind of induction, which is actually a type of deductive reasoning, not inductive reasoning. True inductive reasoning, I'm sure you'll agree, is not mathematically rigorous.

Are you in a philosophy class?

Nope, I am philosophizing on my own free will.

Cool question! I suspect that this kid would be less likely to become extremely good at math - the real-world connections probably provide some insight. I have no evidence whatsoever for this hunch.

Yep, that's what I think. sanabas gave us some evidence of this; it'd be interesting (but probably cruel) to see a full-blown study of this.

 

[Mise]

So, since there doesn't seem to really be a good definition out there, I am going to have to agree with Petek:

I think this is the best possible answer. Like you said, Petek, in the world of mathematics, there must be undefined primitive concepts just as there must be unproven axioms, and I guess "number" is a perfect place to start. And, like you said, if there is actually a suitable definition of number, it'll inevitably require some other primitive concept; for example, Erik's definition leaves "set" as a primitive concept. So yay, you win.


...I think that settles problem #1, but there's still the other half of the mystery (which is perhaps a completely separate mystery, and has nothing to do with what a number is---or maybe it does?), which arises from the fact that mathematics is paradoxically both perfectly concrete and perfectly abstract. Its truths are a priori, yet they can elucidate reality. As Birdjaguar pointed out, mathematics rose out of the need to count things, and that is still the way that (almost?) everyone on this planet is first exposed to math. Some intellectuals even go so far as to dismiss the idea that mathematical truth is in its own little a priori world: in the words of the mathematician Vladimir Arnold, "Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap." (See here.) Could he be right?

One way to think about this is to look directly at the axioms of math. Are the axioms just arbitrary starting points? Or is there a good reason to use them---perhaps the axioms are actually true?

Most people probably don't think it's much of a mystery that the world can be described mathematically; they just take it as a given, as the natural order of things: obviously physics and math are connected; that's just how it is! But there are certainly people who marvel(ed) at the fact:

Albert Einstein: "How can it be that mathematics, being after all product of human thought which is independent of experience, is so admirably appropriate to the objects of realtiy?"

Richard Feynman: "Mathematics is not real, but it feels real. Where is this place?"

Now, if these two brilliant theoretical physicists think there's a mystery, there damn well is a mystery! For more on it, see here. Earlier, I brought up a thought experiment, which I don't think anyone commented on: Imagine we lived in a different universe. (Of course, if the universe were different, we would be different, but just ignore that.) Now, is it possible that, living in that universe, we might be unable to describe its properties mathematically? Or would we always be able to use math to describe the universe, no matter what it's like? Is "using math" simply a matter of finding patterns, and we humans could find patterns in anything---and so the connection between math and reality is, after all, unremarkable?

I can't imagine a universe in which one "thing" plus another "thing" isn't two "things". Unless we make a force which magically creates a third "thing" when a second "thing" is placed next to the first "thing". But then we just have another predictable set of rules. The only way that I can see where mathematics would never be applicable to reality is if the universe is not predictable or based on a certain fixed set of rules, or where the rules change in an arbitrary, unpredictable way.

 

[WillJ]

I can't imagine a universe in which one "thing" plus another "thing" isn't two "things". Unless we make a force which magically creates a third "thing" when a second "thing" is placed next to the first "thing". But then we just have another predictable set of rules. The only way that I can see where mathematics would never be applicable to reality is if the universe is not predictable or based on a certain fixed set of rules, or where the rules change in an arbitrary, unpredictable way.

Yeah, that's what I was thinking. So the mystery is really just that the unvierse is predictable (which is remarkable, if you think about it), which makes it a question of philosophy of physics, rather than philosophy of mathematics.

 

[Abaddon]

The second alphabet.