*Geometry. Courtesy of Wikipedia.*

Let’s start with geometry, although we’ll segue into math in general.

When people think of geometry, they often think of points, lines and so on, so let’s start there. Points and lines are in a relationship, right? For instance, take these two statements:

- Two distinct points define a line.
- Given a line with two distinct points, another point can be found between them.

So a line is defined by points and points are fundamental (no definition is given for them). Now since we already knew what points and lines were, the above is stating the obvious. But what happens if we treat a line as being TOTALLY defined by the above? That is, we pretend we never heard of lines or points, and the above says it all?

To make this clear, let’s replace “point” with X and “line” with Y. This way, we’ll (hopefully) remove the associations we had with points and lines. Now our rules become:

- Two distinct X’s define a Y.
- Given a Y with two distinct X’s, another X can be found between them.

What stands out now is the relationship among the terms; the terms themselves are just place-holders. Now the rules define a relationship.

In fact, looking at these new rules, I can’t help but wonder what I can use for X and Y (other than points and lines) that would satisfy these rules. Well, how about a number for X and a pair of numbers for Y? Then I have:

- Two distinct numbers defines a pair of numbers.
- Given a pair of numbers with two distinct numbers, another number can be found between them.

It’s clear the above works, but let’s do an example. Let {5,10} be the pair of numbers. Can I find a number between them? Sure, how about 6? What’s more, since #2 doesn’t tell me I can only do this once, I can repeat this process with 6 (e.g.: {5,6}) and of course I can find one between that pair: 5.1. I can keep doing this to my heart’s content.

This brings up something else: I must not read anything more into the rules, for doing so weakens the system. For instance, since #2 said nothing about repeated applications, there’s no reason not to repeatedly apply it. It’s this repeated application that gives me a metaphor for rational numbers! Less is more, and getting out of habit of reading my assumptions into things does wonders for my thinking in general.

Back to the point; the above was an example of a (simple) axiomatic system. Axiomatic systems are sets of rules from which (and only from which) we deduce consequences. We leave out our assumptions and even the real world, and try to make the rules our entire world.

See if you recognize this:

- All men are mortal
- Socrates is a man

Using 1 & 2 and the rules of logic, you can then conclude:

* Therefore, Socrates is mortal*

Although the above is known as a syllogism, it is very closely related to an axiomatic system. We lay out the rules and draw conclusions. In fact, I’d consider it an axiomatic system where…

* Premises = Axioms*

* Arguments = Proofs*

* Conclusion = Theorems*

And of course both systems have derivation rules — what makes for valid/sound arguments, rules that are very formal and syntactical.

One difference here may be in how one treats premises vs. axioms. Depending on who you ask, axioms can be self-evident or just definitions, with no claims to truth (one simply deduces the consequences IF the axioms were true). In arguments, one may expect premises to be true and even reject premises of an argument.

Although we started with geometry, we’ve drifted onto math. Math can be treated as an axiomatic system. If so, this makes math the study of tautologies. This isn’t as vacuous as it sounds; while tautologies are things that are true by definition, it doesn’t mean they’re obvious. A tautology may take a very long time and a huge amount of work to tease out, and the results can be very surprising.

When we remove the meanings from terms and study the relationships, are we studying our thought or perceptual processes? It’s an odd thing, given how often we can be surprised by the concepts we invented! It’s almost like writing a story and being surprised at how it’s developing. Such is the adventure of exploring the human mind.

When we explore relationships, we are exploring the ones we notice, which are the ones we value. That I notice a commonality among 2 apples, 2 oranges and 2 cars (the “2”) seems to indicate that I’m predisposed to notice amounts, even if not precisely.

What’s more, this ability to find commonality among disparate situations seems to be an innate skill, as evidenced by our analogies and idioms. For instance, what if I describe a problem as multi-layered?:

*A situation with many delicious layers. Courtesy of Wikipedia*

I know what layers look like (say in a cake), and when I notice a problem with multiple sub-problems, I recognize that outside of the differences between the cake and the problem (indeed, between a visual and mental situation), there’s a shared structure or relationship of the parts of the situation. The parts of the problem relate in a similar way to the parts of the cake.

Is this ability to abstract the irrelevant details to find the structural similarities the driving force behind metaphor, math and language?

Some people have asked why mathematics was so useful. Perhaps this is why. Maybe mathematics by definition studies the way we perceive things. Perhaps mathematics is a very precise, very sophisticated use of metaphor.

**Further Reading**

The World of Mathematics: Volume 3. You needn’t read this whole thing or in order; it’s a series of articles covering a range of topics, so read what interests you. Articles in Parts XI and XII are especially pertinent.

A Portion of Euclid’s elements. Euclid is famous for his attempt to axiomatize geometry, and while his attempt was flawed, it succeeded enough to be considered the model of logic and an inspiration to many. Check out some of his proofs and how he derives them from his axioms. Is he assuming anything? Can you find other objects (models) that can fulfill his axioms? Do the proofs of those models hold? If not, is the flaw in your model, reasoning, or Euclid?

Where Mathematics Comes From: A book about how mathematics arises from the mind. Metaphor plays a role in mathematical thinking here. One of the authors also wrote the next book on this list…

Women, Fire and Dangerous Things. All about how central metaphor and categories are to human thought. Yes, metaphors are not just things we use for the occasional expression, but an integral part in how we view the world.

I loved your way into the subject, generalising from Euclid. A very thought provoking tour around the notions of axioms and categories. It seems relevant to the realism/formalism debate about mathematics – whether we discover it or invent it. I’d be interested to know what your view is.

Thank you! I’ve been trying to find a way to work from Euclid and show the axiomatic system in a readable fashion, so I’m really delighted you liked this and saw what I was trying to do!

I also love that you brought up the realism/formalism debate because I’ve been thinking about that lately, which was one of the inspirations for this post.

My views combine formalism, structuralism and psychologism. I think math is the structure of our observations, and formalism either expresses this structure by its very syntax, or gives us a rigorous way of focusing on the structure by stripping meaning from the terms.

Speaking of which, have you looked at Church Numerals? They are an interesting perspective on the nature of numbers, and the fact that they can be programmed is a huge point in their favor.

What are your thoughts?

I lean instinctively towards realism, but have doubts. Interesting that many leading mathematicians are realists. Is there any sense in sending out e.g pulses of prime numbers in radio waves for putative, unknowable aliens to pick up?

I looked up Church Numerals, but haven’t properly got my head around them, not being any sort of mathematician myself. But I see we are in Gödel / Turing territory, which I have always found fascinating.

Good question… if math is the indirect study of our concepts, then it wouldn’t make sense to expect aliens to pick up on prime numbers, unless they recognize it as an artificial pattern (but not its significance).

Church Numerals treat numbers as functions that take other functions as arguments and call that function that many times. So 0 = no calls, 1 = a single call and so on. The nifty thing is that since the numbers are functions themselves, they can be passed to other numbers. The way you pass them automatically produces addition and multiplication. Further, since those numbers produced are also functions, they too can accept other numbers, get them passed in, and so on. So starting with 0 and 1, you can derive all the other natural numbers.

Yeah, Turing/Church/Curry/Post/Godel territory is awesome stuff. I’m still trying to wrap my head around Godel!

Metaphor is one of my favourite things to think about. Perhaps we understand math by metaphorical thinking, but math itself may not be a metaphor, and I think that may be what you have said. As Gertrude Stein said, “A rose is a rose is a rose is a rose.”

Great to hear from you! I was wondering where you were, and realized my reader didn’t show me as following you. Problem corrected.

Time for some reading 🙂

Math could be a metaphor, parts of it could be a metaphor, it could use metaphor, metaphor may belie it, or maybe a mechanism belies both. How’s that for a waffling answer?

I vaguely remember reading a book that made a somewhat similar argument. I think I might have brought it up before, but the title escapes me.

Meanwhile, a search for Math & Metaphor came up with this. It looks promising, but I only skimmed it:

http://www.mathunion.org/ICM/ICM1990.2/Main/icm1990.2.1665.1672.ocr.pdf

” while tautologies are things that are true by definition, it doesn’t mean they’re obvious.”

Very well said.

Thanks!