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Tue, 05 Sep 2023

Math SE report 2023-06: funky-looking Hasse diagrams, and what is a polynomial anyway?

Is !!x^4-x^4 = 0!! a fourth-degree equation?

This is actually a really good question! (You can tell because it's quick to ask and complicated to answer.) It goes back to a very fundamental problem that beginners in mathematics, which is that there is a difference between an object's true nature and the way it happens to be written down. And often these problems are compounded because there is no way to talk about the object except by referring to how it is written down.

OP says:

The best definition I could find for the degree of an equation is the following:

The highest power of the unknown term whose coefficient isn't zero in a given equation

And they are bothered by this, and rightly so. I was almost derailed at this point into writing an article about what an equation is, but I'm going to put it off for another day, because I think to get to this person's question what we really need to do is to say what a polynomial is.

One way is to describe it as an expression in a certain form, but this is a bit roundabout. It's like describing a rational number as an expression of the form !!\frac n d!! where !!n!! and !!d!! are relatively prime integers. Under this sort of definition, !!x^4-x^4!! isn't a polynomial at all, because it's not an expression of the correct form.

But I think the right way to define a polynomial is that it's an element of the free ring over some ring !!C!! of coefficients. This leaves completely open the question of how a polynomial is written, or what it looks like. It becomes a theorem that polynomials are in one-to-one correspondence with finite sequences of elements of !!C!!. Then we can say that the “degree” of a polynomial is one less than the length of the corresponding finite sequence, or something like that.

[ Sometimes we make an exception for the zero polynomial and say its degree is !!-\infty!!, to preserve the law !!\operatorname{deg}(pq) = \operatorname{deg}(p)+\operatorname{deg}(q)!!.) ]

In this view the zero polynomial is simply the zero element of the ring. The polynomial called “!!x^4!!” is the fourth power of the free element !!x!!.

Since the polynomials are elements of a ring, addition, subtraction, and multiplication come along automatically, and we can discuss the value of the expression !!x^4-x^4!!, which by the usual properties of !!-!! is also the zero polynomial.

Anyway that all is pretty much what I said:

!!x^4-x^4!! is just a way to write the polynomial !!0!!, which is not a fourth-degree polynomial. Similarly !!x^5+x^4-x^5!! is not a fifth-degree polynomial.

There's an underlying reality here, the abstract elements of the ring !!R[x]!!. And then there's a representation theorem, which is that elements of !!R[x]!! are in one-to-one correspondence with finite sequences of elements of !!R!!. The ring laws give us ways to normalize arbitrary expressions involving polynomials. And then there's also the important functor that turns every polynomial ring into a ring of functions, turning the polynomial !!x^4!! into the function !!x\mapsto x^4!!.

This kind of abstract approach isn't usually explained in secondary or tertiary education, and I'm not sure it would be an improvement if it were. (You'd have to explain rings, for one thing.) But the main conceptual point is that there is a difference between the thing itself (say, !!x^4!!) and the way we happen to write the thing (say, !!x^5+x^4-x^5!!), and some properties are properties of the thing itself and others are properties of expressions, of the way the thing has been written. The degree of a polynomial is a property of the thing itself, not of the way it happens to be written, so both of those expressions are ways to write the same polynomial, which is fourth-degree, regardless of the fact that in one of them, “the highest power of the unknown term whose coefficient isn't zero” is five.

There is one example of this abstraction that everyone learns in childhood, rational numbers. I lean hard on this example, because most people know it pretty well, even if they don't realize it yet. !!\frac15!! and !!\frac6{30}!! are the same thing, written in two different ways. Mathematicians will, without even thinking about it, speak of the numerator of a rational number, and without batting an eyelash will say that the numerator of the rational number !!\frac{6}{30}!! is !!1!!. The fraction !!\frac6{30}!! is a mere notation that represents a rational number, in this case the rational number !!\frac15!!, and this rational number has a numerator of !!1!!.

Beginning (and intermediate) computer programmers also have this issue, that the thing itself, usually some embedding of data in the computer's memory, may be serialized into a string in multiple ways. There's a string that represents the thing, and then there's the thing itself, but it's hard to talk about the thing itself because anything you can write is a string. I wish this were made explicit in CS education. Computer programmers who don't pick up on this crucial point, at least on an intuitive level, do not progress past intermediate.

What are the names given to statements that can be true or false?

I think I totally flubbed this one. OP is really concerned with open and closed formulas. For example, “!!x > 2!!” is true, or false, depending on the value of !!x!!. And OP astutely noted that while “!!x>4 \to x> 2!!” is always true, its meaning still depends on the value of !!x!!.

I did get around to explaining that part of the issue, eventually. The crucial point, which is that there are formulas which may have free variables and then there are statements which do not, is buried at the end after a lot of barely-relevant blather about Quinian quasiquotation. What was I thinking? Perhaps I should go back and shorten the answer to just the relevant part.

How does one identify the weakest preconditions in Hoare triples?

I wrote a detailed explanation of how one identifies weakest preconditions in Hoare triples, before realizing that actually OP understood this perfectly, and their confusion was because their book wrote “{x≠1}” where it should have had “{x≠-1}”.


Artifacts of mathematical logic

This was fun. OP wants actual physical artifacts that embody concepts from mathematical logic, the way one models of the platonic solids embody concepts from solid geometry.

I couldn't think of anything good, but then Michael Weiss brought up Lewis Carroll's Game of Logic. This reminded me that Martin Gardner had written a book about embodiments of simple logic, including the Carroll one, so I mentioned that. It's a fun book. Check out the account of Ramon Llull, who missed being canonized because his martyrdom looked a bit too much like FAFO.

I find this answer a little unsatisfying though. The logic machines in Gardner's book do only a little boolean algebra, or maybe at best a tiny bit of predicate logic. But I'd love to see a physical system that somehow encapsulated sequent calculus or natural deduction or something like that. Wouldn't it be cool to have blocks stuck together with magnets or physical couplings, and if you could physically detach the !!B!! from !!A\to B!! only if you already had an assemblage that matched !!A!! exactly? I have no idea how you'd do it. Maybe a linear logic model would be more feasible: once you used !!A!! with !!A\to B!! to get !!B!!, you wouldn't be able to use either one again.

We need some genius to come and invent some astonishing mechanism that formerly seemed impossible. I wonder if Ernő Rubik is available?

Joachim Breitner's Incredible Proof Machine is a fun thing along these lines, but it's not at all an artifact.

Is there a name for this refinement of the subset ordering?

This was my question. I've never seen this ordering elsewhere, but it has a simple description. We start with a totally ordered finite set !!S!!. Then if !!A!! and !!B!! are subsets of !!S!!, we deem !!A \preceq B!! if there is an injective mapping !!f:A\to B!! where !!a \le f(a)!! for each !!a\in A!!.

So for example, if !!S!! has three elements !!a\lt b\lt c!! then the ordering I want, on !!2^S!!, has this Hasse diagram:

At far left, least, is the empty set.  Then, in a
line to the right, the set {a}, then {b}, then both {ab} and {c}, then
right of both of these is {ac}, then {bc}, then S={abc}

!!\{b\}\prec\{a,b\}!! because we can match the !!b!!'s. And !!\{a,b\}\prec \{a, c\}!! because we can match !!a!! with !!a!! and !!b!! with !!c!!. But !!\{c\}\not\prec\{a, b\}!! because we can't match !!c!! with either !!a!! or with !!b!!, and !!\{a,b\}\not\prec\{c\}!! because, while we can match either of !!a!! or !!b!! with !!c!!, we aren't allowed to match both of them with !!c!!.

Here's the corresponding Hasse diagram for !!|S|=4!!:

A more complicated
Hasse diagram with 16 nodes, one for each subset of {a,b,c,d}.

Maybe a better way to describe this is: the bottom element is !!\varnothing!!. To go up the lattice one step, you either increment one of the elements of the current set, or you insert an !!a!! if there isn't one already. So from !!\{b,d\}!! you can either increment the !!b!! to move up to !!\{c, d\}!! or you can insert an !!a!! to move up to !!\{a, b, d\}!!.

This ordering comes up in connection with a problem I've thought about a lot: Say you have a number !!N!! and you want to find !!AB=N!! with !!A!! and !!B!! as close together as possible. Even if you have the prime factorization of !!N!! available, it's not usually clear what the answer is. (In general it's NP-hard.)

If !!N!! is the product of two primes, !!N=p_1p_2!! the answer is obvious. And if !!N!! is a product of three primes !!N =p_1p_2p_3!! there is a definitive answer. Without loss of generality, !!p_1 ≤ p_2 ≤ p_3!!, and the answer is simply that !!A=p_1p_2, B=p_3!! is always optimal.

But if !!N =p_1p_2p_3p_4!! it can go two different ways. Assuming !!p_1 ≤ p_2 ≤ p_3 ≤ p_4!!, it usually turns out that the optimal solution is !!A=p_1p_4, B=p_2p_3!!. But sometimes the optimal solution is !!A=p_1p_2p_3, B=p_4!!. These are the only two possibilities.

Which ways of splitting the prime factors might be optimal relates to those Hasse diagrams above. The possibly-optimal splits between !!A!! and !!B!! correspond to nodes that are just at the boundary of the left and right halves of the diagram.

Nobody had an answer for what this order was called, so I could not look it up. This is OK, I will figure it all out eventually.

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