My Math Forum How are integers a subset of R^2?

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December 28th, 2016, 06:45 PM   #21
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Math majors just pick up the idea of identifying things with their isomorphic copies. It's a common pattern you absorb over time.

In the process, you get a sense that math isn't "based on" set theory, as people have heard, but rather that set theory is a tool that helps you to do math. You don't let the fine points of set theory get in your way.

A formal approach to this point of view is Category theory. It started out in the 1940s and has now spread throughout many different areas of math and even applied areas like physics and computer science. In Category theory, you're concerned with the relationships between objects and never with the internal structure of the objects themselves. So the distinction between the integers as integers and the integers as real numbers never comes up because there's only one object that acts like the integers.

https://en.wikipedia.org/wiki/Category_theory

This is what I associated with structuralism but I should admit that this is a very vague connection. In fact, there's a Philosophy.StackExchange thread asking whether there's a connection between structuralism and Category theory, and the consensus is "probably not". So maybe I should go light on the bit about structuralism.

philosophy of mathematics - Is there any connection between Structuralism and Category Theory? - Philosophy Stack Exchange

But there's no question that a large chunk of modern math takes place in the framework of Category theory and not set theory.

Another interesting alternative is Homotopy Type theory (HoTT). HoTT relates to intuitionist mathematics and the idea of the rejection of the excluded middle. These ideas are gaining currency via computer science. In CS theory, we have sets of natural numbers such that neither the set nor its complement are computable. So if you believe that everything meaningful is computable then you can't believe in classical math. HoTT is intimately connected with developments in computer-aided proof verification. It's a few years old and becoming popular.

https://en.wikipedia.org/wiki/Homotopy_type_theory

Quote:
 Originally Posted by Joppy What sort of consequences (if any) are there if we incorrectly view certain problems in a structuralist or formalist perspective (if this is correct usage of the terms)?
I don't think there are consequences, just different ways of looking at things. The type theorists would probably say the subset problem is some sort of type error. I don't really know much about type theory.

In Category theory (which I associate with structuralism despite the absence of much Web evidence) there is no problem, because there are no two things that are isomorphic. There's only the one thing, the structure itself, and not its set-theoretic implementation. I'm not explaining that very well, but Category theory can't be described in a few words.

Perhaps an analogy is if a sociologist is looking at the relationship between two people. They wouldn't be concerned about our internal organs, but only the relationships and transactions between the people. Category theory is like that.

Quote:
 Originally Posted by Joppy Is this in anyway similar to the platonic and formalist perspectives?
I suppose. I should mention that I am not authoritative in matters of philosophy. But if we have two different sets and both claim to be the integers, we do have a problem for the formalists. If the integers can be represented in two different ways, then the integers are the abstract pattern the two sets have in common; and not the sets themselves.

So if the integers aren't the set that represents them, what are they? We're back to square zero, ontologically speaking.

Last edited by skipjack; December 30th, 2016 at 07:38 PM.

December 28th, 2016, 09:05 PM   #22
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Sorry i seem to have interpreted parts of post #17 incorrectly. In any case, thanks for the response, and links.

Quote:
 Originally Posted by Maschke In Category theory you're concerned with the relationships between objects and never with the internal structure of the objects themselves.
I can make sense of this given my (brief) experience in object orientated programming (OOP). This makes up the basis for the fundamental principles of OOP.

Quote:
 Originally Posted by Maschke I should mention that I am not authoritative in matters of philosophy.
Neither am i, but we can engage in some not-so-philosophical banter.

Quote:
 Originally Posted by Maschke But if we have two different sets and both claim to be the integers, we do have a problem for the formalists.
Like how the set $S$ has a subset of itself, $S'$? i.e., the two sets are defined differently, but are ultimately the same. Is this your meaning of 'different' sets claiming to be the same thing?

I'm guessing no since this case isn't a problem for formalists. In which case, there is no way to represent the integers in another way?

Quote:
 Originally Posted by Maschke So if the integers aren't the set that represents them, what are they?
Perhaps it is the abstract pattern that differentiates the two (or perhaps more) sets of integers? Which would complement category theory in it's focus of relationships between entities and not the entity itself, as you have said.

Quote:
 Originally Posted by Maschke We're back to square zero, ontologically speaking.
As is usually the case in such matters .

December 29th, 2016, 10:16 PM   #23
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Quote:
 Originally Posted by Joppy Sorry i seem to have interpreted parts of post #17 incorrectly. In any case, thanks for the response, and links.
If it's still unclear let me know.

Quote:
 Originally Posted by Joppy I can make sense of this given my (brief) experience in object orientated programming (OOP). This makes up the basis for the fundamental principles of OOP.
Yes that's an excellent analogy. In OOP we structure an application as a collection of interacting objects whose internal structure is opaque to the other objects.

I think an even better analogy is that Category theory is design patterns for mathematicians. Design patterms came out of object theory once people realized they were using the same OO designs in many different applications. Category theory describes and formalizes common patterns among mathematical objects.

For example in set theory we have the idea of the Cartesian product of two sets, which consists of the collection of ordered pairs, one from each set. In Group theory there's a construction called the direct product of groups, which builds on the Cartesian product but adds a binary operation on the ordered pairs. In ring theory there are product rings. In topology there are product topoologies. As a familiar example, $\mathbb R^2$ is the vector space product of $\mathbb R$ with itself.

Category theory unifies the notion of the product of two mathematical objects by clarifying the relationship between the product and its factors. This is done via an arrow diagram known as a commutative diagram.

These two Wiki pages give the flavor of what this looks like. The first time you see diagrams they're not like any other math you ever saw. This is a revolution in math that's been going on since the 1940's and has now taken over algebra and geometry entirely. A lot of math these days is all about chasing commutative diagrams.

This page on the categorical product will really give you a good sense of the flavor of this approach.

https://en.wikipedia.org/wiki/Product_(category_theory)

Also see

https://en.wikipedia.org/wiki/Commutative_diagram

Quote:
 Originally Posted by Joppy Like how the set $S$ has a subset of itself, $S'$? i.e., the two sets are defined differently, but are ultimately the same. Is this your meaning of 'different' sets claiming to be the same thing?
Not exactly. I have an example below.

Quote:
 Originally Posted by Joppy I'm guessing no since this case isn't a problem for formalists. In which case, there is no way to represent the integers in another way?
I think it's a terrible problem for the formalists because it's clear that mathematical objects can not possibly be the sets that represent them. @ProofOfALifetime's remark about the complex numbers is a great example to illustrate this point.

Quote:
 Originally Posted by ProofOfALifetime Exactly, this is what was throwing me off. But, I learned that according to rudin he regards the set of complex numbers as $R^2$, so remembering that helped me as well. The reason it helped me is because we can write integers as complex numbers with imaginary part zero. (He regards them as ordered pairs so in other words, we make the 2nd coordinate $0$.)
Great example. Suppose some skeptic says, "I'll grant you the real numbers but I just don't believe in the complex numbers."

Then we say Ok, if you grant us $\mathbb R$ then surely you'll grant us ordered pairs $(a, b)$ of reals. Then we define addition on ordered pairs:

* $(a,b) + (x,y) = (a + x, \ b + y)$. That's well defined because we already know how to add real numbers. This is familiar vector addition in the plane.

And we define multiplication of ordered pairs with a seemingly funny-looking rule:

* $(a,b)(x,y) = (ax - by, \ bx + ay)$.

Then we prove that the set of ordered pairs with this addition and multiplication behaves exactly as we expect the complex numbers to behave. For example $(0,1)(0,1) = (-1, 0)$. So from now on we call $a,b$ by the name $a + bi$ and we're in business. [Fun exercise: Prove that ever nonzero complex number has a multiplicative inverse, in other words that $\mathbb C$ is a field].

The skeptic goes away happy because they see that if they believe in the real numbers they must believe in the complex numbers. But a more sophisticated skeptic knows that the ordered pairs construction isn't the only way to do it; so that although ordered pairs model the complex numbers, they can't possibly be the complex numbers.

In undergrad abstract algebra (second semester, so it's not entirely elementary) they do the following construction. Start with $\mathbb R$, which our skeptic has agreed to. Now we all learned about polynomials in high school, so let $\mathbb R[x]$ be the set of all polynomials in one variable with real coefficients. Things like $3x^5 - \pi x^2 + 8$. We all learned how to add and multiply them. They obey a lot of the same rules as the plain old integers. There's a zero object, every polynomial has an additive inverse, multiplication distributes over addition, etc. In the integers we can "mod out" the multiples of $5$ to construct the integers mod $5$. By analogy, we can mod out the multiples of the polynomial $x^2 + 1$. The resulting object is called $\mathbb R[x]/(x^2 + 1)$. And it turns out to be a field isomorphic to ... drum roll ... the complex numbers!

And if we're in linear algebra class, we can even define the complex numbers as the set of $2 \times 2$ matrices of the form

$\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$

where $a$ and $b$ are real numbers, with the usual matrix addition and multiplication.

So now we have a philosophical problem. If we think that the complex numbers "are" ordered pairs, then they can't also be equivalence classes of polynomials or a certain set of matrices.

We are forced to admit that the "actual" complex numbers are not a set-theoretical object at all; rather, they're the pattern expressed by all of these isomorphic structures.

In which case we no longer know what the complex numbers "are." We only know how to build models of them out of pile of tinker toys mathematicians call set theory. [Do people still know what tinker toys are?]

In a sense, set theory is the least important part of the process. The internal structure of ordered pairs or polynomials or matrices is totally irrelevant to the true nature of the complex numbers.

We might almost say that the set theoretical constructions get in the way! They're scaffolding used to construct the building but they are not the building. To see the building we have to get rid of the scaffolding when we're done with it.

It might be said (pushing a point a little) that twentieth century set theory was the scaffolding; and now mathematicians are learning to see what's left after you take the scaffolding away.

Last edited by Maschke; December 29th, 2016 at 10:23 PM.

December 30th, 2016, 04:00 PM   #24
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Hope this helps!

Quote:
 Originally Posted by ProofOfALifetime Exactly, this is what was throwing me off. But, I learned that according to rudin he regards the set of complex numbers as $R^2$, so remembering that helped me as well. The reason it helped me is because we can write integers as complex numbers with imaginary part zero. (He regards them as ordered pairs so in other words, we make the 2nd coordinate $0$.)
I am writing a very very informal proof, but see note at bottom.

Rudin is referring to an interval (a,b).

This is indeed a subset of $R^{1}$, since you could create a one-dimensional line, which clearly exists in one-dimensional space.

Your question, I believe, is how is this a subset of $R^{2}$.

Recall the definition of a subset:

A set $A$ is a subset of another set $B$ if and only if all elements of $A$ are contained in $B$.

This requires that we define our sets.

Let set $A=\{ ...,-2,-1,0,1,2,... \}$, almost like individual x-values, and set $B=\{ ...,(-2,0), (-1,0), ... \}$.

If you drew these out, you could see that on a number line, they are in fact the same positions for all value of $x$.

However, if set B's y-value deviates from $0$, further up or down in $R^{2}$, then you have an issue.

But, that would be checking to see if $B$ is a subset of $A$, which it is not. Rudin also makes this disclaimer at the bottom of the example.

Since every $x$ value in $A$ is contained in the points of $B$ when $y = 0$, $A$ is therefore a subset of $B$.

I have simply restated what I saw you mention regarding complex numbers, however it was unnecessary to relate to that, so I thought I would show it.

Note: I used x and y ( function notation ) for simplicity, but if you would like a formal proof using Set Theory Notation, just let me know. I don't mind writing it out formally as Set Theory is pure fun to me.

December 30th, 2016, 05:38 PM   #25
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Quote:
 Originally Posted by thegrade A set $A$ is a subset of another set $B$ if and only if all elements of $A$ are contained in $B$. This requires that we define our sets. Let set $A=\{ ...,-2,-1,0,1,2,... \}$, almost like individual x-values, and set $B=\{ ...,(-2,0), (-1,0), ... \}$. If you drew these out, you could see that on a number line, they are in fact the same positions for all value of $x$.
I don't mean to be picky here -- (Just kidding. Of course I'm about to be picky!) -- but you have not shown that $\mathbb R^1 \subset \mathbb R^2$. An element of the reals is a number, like $5$. An element of the plane is an ordered pair, like $(5,0)$. As sets, they are not the same. So strictly speaking it is not true that $\mathbb R^1 \subset \mathbb R^2$.

The fact that we can overlay the real line on a particular subset of the plane in such a way that the real number $x$ directly overlays the point on the plane $(x, 0)$, simply says that there's an order-preserving bijection from the reals to a subset of the plane. In fact we can go further and note this bijection preserves addition and multiplication too. So we say that since the plane contains a subset isomorphic to the reals, from now on we'll just call it the reals and say that "the reals are a subset of the plane."

That's what the subject of the thread is about. Strictly speaking, according to the formal definition of subset, the reals are NOT a subset of the plane. If we programmed a computer with the axioms and rules of set theory, and we asked it if the reals are a subset of the plane, the answer would be NO.

When asked if the reals are a subset of the plane, we say YES, while secretly thinking, "Not in terms of set theory, but we can identify the reals with a particular subset of the plane and that's good enough."

The philosophical point is that in spite of set theory's 20th century success as the foundation of math, there's something not quite right about it. It's the isomorphism that's important; and not the specific set-theoretic construction of the line and the plane.

Category theory is one response to this problem. Modern type theory is another.

Last edited by Maschke; December 30th, 2016 at 06:01 PM.

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