How do you show one way equivalences in mathematics?

Question

In the real world, it seems you can often take a set of things and create new things with them. Let's say you want to make bread, for example. Normally, you can create bread by putting together flour, milk, eggs, and a few other ingredients, and cooking.

It would seem, that this could be written as:

$flour+milk+eggs+(other ingredients)=bread$

I believe there is a problem with this though, in that you can't create milk or flour (in any practical way) from bread. It just wouldn't work.

So, in specific to that problem, I'm wondering of there is a way to identify one way equivalences in mathematics. That is $x$ and $y$, such that $x$ can turn into $y$, but $y$ cannot turn into $x$. I guess that can lead to the more complicated issue of causation, which I'm also wondering about.

Answer 1

I'd like to expand upon my comment because this is a very interesting question.

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First, I think the word "causation" is really throwing people off. Causation does have a specific meaning that's been modeled in various ways (I'm thinking statistics and mathematical logic), but none of the ways I know about capture what (I think) you're after.

The link I posted in my comment is to a post about "Resource Convertibility," which really perfectly captures what you've been thinking about, or at least the example you gave. I don't know much about it, but it's fun to think about, so I'll try to say a little bit.

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You've got a very interesting idea that surprisingly intersects with some current research!

Unfortunately, it may turn out that equality has connotations that are inappropriate, as you've already seen:

$$\rm flour\ + milk\ +eggs\ + (other\ ingredients) = bread$$

is indeed something of an unfortunate notation, because equality is symmetric and as you've pointed out, our conversion here is really one way.

Tobias Fritz has evidently been thinking about this, and decided that an inequality was really the way to go: It makes more sense to write

$$\rm flour\ + milk\ +eggs\ + (other\ ingredients) \ge bread$$

and think of this statement as something like "having flour, milk, eggs, and other stuff is at least as good as having bread" (highly paraphrased from link above). It's also probably best to avoid causation and speak strictly about the ability to convert the things on the left to the things on the right (as the act of bringing milk and eggs together certainly doesn't cause bread to form of its own volition!).

The key features of his formulation are that

• You have some means of "comparing" objects, using $\ge$.
  • Everything is comparable to itself (i.e., $\rm bread \ge bread$).
  • Comparisons are transitive. For example if $\rm bread\ ingredients \ge bread$, and $\rm bread \ge toast$, then $\rm bread\ ingredients \ge toast$ (just bake the ingredients into bread, then slice and toast!).
  • Finally, antisymmetry (think $x \ge 3$ and $3 \ge x$ means $x = 3$) is used as a means to decide what resources are equivalent. We have, for example (by visiting one's favorite financial institution and trading cash for different cash),

$$\rm five\ \$1\ bills \ge one\ $5\ bill \qquad and \qquad one\ $5\ bill \ge five\ \$1\ bills.$$

• We also have the ability to add objects, as in the $\rm flour + milk + \ldots$ example (but the $+$ really just lets us build a shopping cart from what I can tell; adding doesn't convert anything, that's all encapsulated in $\ge$).

With these abilities, you get to say you're studying fancy things called ordered commutative monoids, and you can read much much more in the series of posts linked above, as well as the paper that ensued (there's a lot of fancy notation and theorems, but there's some value to skimming at least portions of the paper, if you find the blog series interesting enough).

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Has any of this turned out to be useful, from a practical standpoint? I have no idea! But John Baez (the person running the blog) as been involved in, and popularized, efforts to build a framework to talk and think about network theory: Chemical reactions (think $H + O \ge H_2O$), birth-death processes, resource conversion, etc. It turns out that classical mathematics can only say so much about these subjects.

This perspective on Resource Convertibility is just one of many efforts to find a good framework.

Answer 2

I think the mathematical concept you want is implication.

For example, think about mathematical equations that only work sometimes. How about $x^2=x$. If you plug in a random value of $x$, that equation isn't true. But it happens to work when $x=1$. Written mathematically,

$$ x = 1 \quad \Longrightarrow \quad x^2 = x$$

The symbol $\Longrightarrow$ is read as implies, or as the then part of an if-then statement. You can also think about it as meaning mathematical causation -- e.g. the fact that $x=1$ causes the equation $x^2=x$ to be true. This arrow doesn't always go both ways. If it's true that $x^2=x$, that doesn't imply that $x=1$. (It could instead be true that $x=0$.)

Warning: there are some problems with this perspective, namely that in mathematical logic, any false statement implies any statement you want. This is worth thinking about some more!

Answer 3

I would say the best mathematical setting to represent physical causation are dynamical systems:

• You have a global "state" $x(t)$ (a vector with many coordinates) with flour, milk and eggs represented by set of variables changing in the time variabe $t$,
• you have physical laws represented by a rule linking $x(t)$ to $x(t+dt)$ (maybe a differential equation),
• you have a a "macro"-state $X(t)$ that is represented by a set of "micro"-state $x(t)$ that we identify (for example there are several micro-state that we would call "bread" and several micro-states that we would call "unmixed flour, milk and eggs"),
• you have causation if the macro-state $A$ (in any given possible micro-state configuration) always evolves to the macro-state $B$ (in some micro-state configuration), then you would say that "$A$ causes $B$".

Answer 4

If you are interested in causality, then the book "Causality: Models, Reasoning, and Inference" by Judea Pearl should be of interest to you. It's a little difficult to judge from your question what you are actually after, but my guess is you will find plenty of illumination in that book.