B Is everything in math either an axiom or a theorem?

[pinball1970]

The only way to interpret that question becomes a familiar debate about whether numbers (and other mathematical concepts) were discovered or invented. You are suggesting that human beings developed language to describe something that already existed.

My take on that:

If we interpret "exist" to mean something that exists in a purely physical sense (like my coffee cup) then I can verify the existence of examples where (the current) concept of numbers models many of the physical properties well. Finding such examples requires not only finding examples of the noun number, but it also requires giving a physical interpretation to what we mean by "=", "+" etc.

I don't regard numbers as having an existence that physically causes 3+2 = 5. There are physical situations where this doesn't work (like computing the final volume after two volumes of different chemicals are added to the same flask). One can always object to such examples by saying that they are not what is meant by 3+2 = 5. This is a convenient type of a argument. It merely says that situations where a property of numbers works prove the physical existence of that property and situations where the property doesn't work are automatically disqualified from consideration.

No I don't mean in the physical sense. I have to use a situation to explain it however. (This is an outsider/lay view so it will sound simplistic but I would like to ask this)
An alien species sends a probe to our solar system and finds planets orbiting a star.
There is a quantity of planets and properties of that quantity that exist.
You can add other quantities to reach the total, you can times a quantity by another quantity to reach the total. (They like pluto)

Take away the planets and those relationships still exist, no matter what you call them.
The chemical reaction I would object that you are changing the quantities as you are adding them, losing some volume as gas or precipitate.
It works if you just consider two generic volumes x/2 and add them you end up with volume x.
It works if you completely remove all physical reality and consider two abstract objects x/2 and add them you ALWAYS end up with x.
Does that relationship just exist?

 

[Stephen Tashi]

It works if you just consider two generic volumes x/2 and add them you end up with volume x.

That doesn't necessarily work if you "add" two subcritical volumes of U-235 in a particular way. It may not work if you add two volumes of water that are at different temperatures.

It works if you completely remove all physical reality and consider two abstract objects x/2 and add them you ALWAYS end up with x.

I don't know what it means to add abstract objects. Perhaps you are thinking about computing the cardinality of the union of two sets. Depending on how we interpret "add", it may or may not work with physical objects. It seems to me that you are defining abstract objects to be things x that have the property that x/2 + x/2 = x. So the relationship exists just by your definition

I agree that many physical situations exist where the abstract concept of addition can be interpreted and applied to make a sucessfull prediction. So the abstract concept of numbers and addition exists in sense that there exist physical examples where it can be applied.

It is human beings who recognized the similarities in these situations and created the abstract concept of numbers and addition. This improves the efficiency of thought - employ one abstract pattern that can be applied in many situations instead of treating each situation separately.

 

[nuuskur]

There are statements. Classically, every statement is either true or false (but not both). The end.
You can call them statements, propositions, theorems, doesn't matter.

Other than that - either axiom or theorem is sufficiently accurate. Theorem meaning that it requires proof. An axiom is just regarded as a true statement.

 

[pinball1970]

That doesn't necessarily work if you "add" two subcritical volumes of U-235 in a particular way. It may not work if you add two volumes of water that are at different temperatures.I don't know what it means to add abstract objects. Perhaps you are thinking about computing the cardinality of the union of two sets. Depending on how we interpret "add", it may or may not work with physical objects. It seems to me that you are defining abstract objects to be things x that have the property that x/2 + x/2 = x. So the relationship exists just by your definition

I agree that many physical situations exist where the abstract concept of addition can be interpreted and applied to make a sucessfull prediction. So the abstract concept of numbers and addition exists in sense that there exist physical examples where it can be applied.

It is human beings who recognized the similarities in these situations and created the abstract concept of numbers and addition. This improves the efficiency of thought - employ one abstract pattern that can be applied in many situations instead of treating each situation separately.

I had a look around and from what I have read, numbers are a human construct. Mathematics too.
I do not understand that.
This is straying from the OP so I may risk a thread instead.
Frustrating because I do not know enough about the structure and theory to get to the crux of what I mean.
I may have to mail a mentor first to make sure my post makes sense and does not break the rules!

 

[PeroK]

I had a look around and from what I have read, numbers are a human construct. Mathematics too.
I do not understand that.
This is straying from the OP so I may risk a thread instead.
Frustrating because I do not know enough about the structure and theory to get to the crux of what I mean.
I may have to mail a mentor first to make sure my post makes sense and does not break the rules!

Bring me a number! The whole numbers are an idea. They don't exist independently in nature.

You could use a pile of marked stones to count. That is, in fact, the origin of the term calculus. The leap is to invent an abstract set of things, called numbers, that are free of any single physical representation.

That was a significant breakthrough in the history of mathematics.

 

[pinball1970]

Bring me a number! The whole numbers are an idea. They don't exist independently in nature.

You could use a pile of marked stones to count. That is, in fact, the origin of the term calculus. The leap is to invent an abstract set of things, called numbers, that are free of any single physical representation.

That was a significant breakthrough in the history of mathematics.

I am saying the opposite, the structure exists without us. Regardless of the need for numbers to describe anything, that is just a bonus.
A being in Andromeda will arrive at the same results.

I am happy to be wrong but I think I need more meat to the bones before I take you on.
Let me check out number theory and set theory, possibly the philosophy of maths.
I could be a while...

 

[PeroK]

I am saying the opposite, the structure exists without us.

That probably depends on your definition of "exists". The number five? The set of $n \times n$ invertible matrices? The quaternions? The more advanced the mathematics, the less convinced I am of its independent existence, whatever that means.

 

[pinball1970]

That probably depends on your definition of "exists". The number five? The set of $n \times n$ invertible matrices? The quaternions? The more advanced the mathematics, the less convinced I am of its independent existence, whatever that means.

Fair enough,I know the platform I am claiming from. It is shaky ground.
Let me get back.

 

[pbuk]

I am saying the opposite, the structure exists without us.

That is a matter of philosophy, not maths.

It seems that as long as you stay away from self-reference, you should be fine.

That gets quite tricky: induction is a key principal in mathematics and is a close cousin of self-reference.

I think it would be a good idea to clear up some of the abstract talking in this thread by actually looking at some axioms. Let's look at the Peano Axioms (I'm going to base this on the construction used by MathWorld).

1. Zero is a number.
2. If $a$ is a number, the successor of $a$ is a number.
3. Zero is not the successor of any number.
4. Two numbers of which the successors are equal are themselves equal.
5. If a set $S$ of numbers contains zero and also the successor of every number in $S$, then every number is in $S$.

These axioms construct, completely artifically, the set of natural numbers, and when most mathematicians in the past 100 years talk about natural numbers this (or something similar) is what they are talking about. Using these (and a few more) we can construct propositions such as $2 + 3 = 5$ and $2 + 4 = 5$ and we can attempt to prove them. We find that we can prove that $2 + 3 = 5$ is always true and so we say it is a theorem of Peano Arithmetic, whereas we can prove that $2 + 4 = 5$ is always false and so we say it is not a theorem of Peano Arithmetic.

Now we can start again with a different set of axioms (I dedicate these to @Feynstein100 so let's call them the Pbuk-Dedicated axioms ):

1. Unicorns are animals
2. Unicorns have exactly one horn

... from which we can prove theorems such as "if $u$ is a unicorn then $u$ has exactly one horn".

We can also construct the proposition $P$ "if $a$ is an animal and $a$ has exactly one horn then $a$ is a unicorn" and its negation $P' = \neg P$ "if $a$ is an animal and $a$ has exactly one horn then $a$ is not a unicorn" but we find that we cannot prove either $P$ or $\neg{P}$ using the Pbuk-Dedicated axioms and so we say that $P$ is undecidable in Pbuk-Dedicated arithmetic.

So you see the concepts of axioms, propositions, theorems and decidability exist completely independently of any underlying "truth" behind the axioms.

 

[Feynstein100]

So we got a little distracted. In summary: turns out, yes, everything in math is either an axiom or a theorem. There is a third possibility with undecidable statements but that's too complicated. For most purposes, the duality of axiom/theorem holds. I'm glad we sorted that out.
Correction: Every statement in math is either an axiom or theorem. Math also consists of things that aren't statements, namely mathematical objects. I guess the interaction between these objects could be considered statements but that's not really relevant here.

 

[jbriggs444]

everything in math is either an axiom or a theorem. There is a third possibility with undecidable statements but that's too complicated.

If you slice away everything that is "too complicated", many false claims become true.

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

 

[SSequence]

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

This seems like a reasonable view to me.

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The way I see it simply this way. If we have a decidable theory then (assuming consistency) every statement falls exactly into one of the following two categories:
(i) provable in the theory (or a theorem of the theory)
(ii) disproveable in the theory

The statements in category-(ii) that would be disproveable (in the theory) would have their negation as a theorem [theorem of the theory that is].On the other hand, if we have an incomplete theory under consideration [as often happens to be the case] then (assuming consistency) every statement exactly falls into one of the following three categories:
(i) provable in the theory (or a theorem of the theory)
(ii) disproveable in the theory
(iii) independent

And once again the statements in category-(ii) that would be disproveable (in the theory) would have their negation as a theorem [theorem of the theory that is].

 

[Feynstein100]

If you slice away everything that is "too complicated", many false claims become true.

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

Fine. It's not a duality but a trinity: Axiom, Theorem or Undecidable

 

[pinball1970]

If you slice away everything that is "too complicated", many false claims become true.

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

Yourself regarding infinitities and Cantor @PeroK generally and @fresh_42 have been valued over the years. Thank you. Fresh insights are good.

 

[EventHorizon]

Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more?
I feel like we're venturing into Gödel incompleteness territory here but for this discussion, let's keep it simple. Is this duality of axiom/theorem all-encompassing or are there things that lie beyond?
It's also interesting to note that since theorems follow from axioms, it should be possible to write down all axioms and all their subsequent theorems, in a linear timelike/causality-like structure. Is this a coincidence?
And finally, where does the notion of mathematical objects fit into this? For example, you might find out all axioms and theorems related to numbers. But once you introduce the idea of vectors and tensors, which are different mathematical objects, the previous knowledge doesn't apply anymore. Because they are different objects, they will have different properties, meaning different axioms and theorems. Basically, any new mathematical object will have its own set of axioms and theorems. Which raises the question, will there always be some new mathematical object to discover i.e. there are an infinite number of them or will we eventually run out of them?

There are also things like the Navier-Stokes equation where there are no defined "solutions"

 

[Vanadium 50]

There are also things like the Navier-Stokes equation where there are no defined "solutions"

What? Are you guessing again? Please don't do that.

Of course there are solutions. Otherwise it would be fairly useless. What there are not are closed-form solutions involving only elementary functions for arbitrary initial conditions.

If you are referring to the Clay Instiute's Navier-Stokes problem, that's something rather different from what you wrote.

It's aslo kind of an off-topic necropost.

 

[renormalize]

There are also things like the Navier-Stokes equation where there are no defined "solutions"

Do you refer here to the lack of formal proofs of existence, uniqueness and smoothness of solutions of the N-S equation?

 

[Rfael]

you have 'axioms' 'theorems' 'lemmas' and 'corollaries' :D

 

[Svein]

Did we invent the numbers? Give them properties? The nomenclature is ours (Arabic, Roman, Babylonian etc) but did we invent the actual numbers?
Give them properties? Even numbers, primes?
EDIT: I know this is straying away from the OP a little.

Some number systems are better than others in expressing mathemathical conjectures. The Peano axioms are meaningless in roman numerals - leaving aside the fact that 0 does not exist, the notion of a "successor" is not trivial (what is the successor of VIII?)

 

[jbriggs444]

Some number systems are better than others in expressing mathemathical conjectures. The Peano axioms are meaningless in roman numerals - leaving aside the fact that 0 does not exist, the notion of a "successor" is not trivial (what is the successor of VIII?)

I suppose that we could shift our numeral system to a Gray code. Though I doubt it would make constructing the reals from the Peano axioms any easier.

 

[FactChecker]

I've been meaning to ask about that. It seems to me that what Gödel actually discovered is that self-reference is different kind of thing that isn't compatible with normal everyday logic i.e. the pattern of true/false doesn't necessarily apply to self-referential statements. I have yet to see an example of it in a non-self-referential context i.e. a statement that isn't self-referential and not true/false. It seems that as long as you stay away from self-reference, you should be fine.

I think it is important to distinguish between the number of counterexamples that are easily proven to be counterexamples versus the number of counterexamples that exist. There are $\aleph_1$ transcendental numbers, but far fewer proven ones.

 

[Henryk]

I'm certainly not a mathematician, can you give an example of something known to be true but unproven? In my mind, axioms are "given" -- assumed to be true. But they may not "really" be true. Things like flat space.

There is no such thing as not "really" true in mathematics. Maybe the general theory states that the physical space is not actually flat, but that doesn't mean we can't construct a true flat space in mathematics.

 

[Feynstein100]

you have 'axioms' 'theorems' 'lemmas' and 'corollaries' :D

Just when I thought I was done, they pull me back in