Even though most of us use mathematics in our daily lives, few of us ever think about what mathematics is about. We add, subtract and divide routinely, without thinking twice about what we’re doing. We assume math is something that relates to numbers and the relationships between them. We assume wrong.
In this article I want to take a look at the different ways of looking at math and its meaning. I will adopt a pseudo-historical approach: by this I mean I will make the different views unfold like a story in something that may look like history but probably has absolutely nothing to do with actual history. I try to keep my articles reasonably short, so there will be flagrant omissions: for example, I purposely ignore the intuitionist view for two reasons: first, it didn’t fit well with my narrative; second, I am not well-versed in its philosophical underpinnings. So feel free to expand on the topic in the comments below.
Mathematics started indeed being about numbers and how to add them up. It was a practical matter, more than an intellectual one. “You owe me three pigs and a goat.” That was math. The notion got refined as operations got more complex, especially with the appearance of geometry. It quickly became apparent that, applying elementary reasoning techniques (logic) to some set of simple, o
bvious truths, one could get extremely complex and non-obvious results. This lead many to asks themselves: what are we really doing when we do math?
Plato. Didn’t look too cute in this photo.
Probably the Platonic answer is the most popular. According to Plato, we could access mathematical truths because we had experienced them before in the realm of ideas. That is: somewhere, there is such a thing as a perfect triangle, of which every other triangle is but a rough copy. Mathematical statements, then, are statements about these perfect objects which exist outside our current realm of experience.
The Platonic approach was ontologically loaded: it assumed the world to be a certain way. It turns out most mathematicians don’t like the world much. They like what goes on in their head much more. So an answer based on reality was not satisfying to many, who were looking for a much more abstract, mathematically-minded way of looking at things.
A view that fit much better with this desire for abstraction was that of mathematics as the expression of logic. In this sense, mathematics would be nothing but pure logic, applied to certain propositions. Those initial propositions, given without proof, were called axioms. If one sticks to this interpretation, mathematics is the only branch of knowledge that deals with absolute truths, even if the initial axioms are not true. This is so because all that mathematics state is: “if these series of axioms is true, then so are these other statements.”
An example will clarify things. Imagine my starting axioms are:
“I have four arms.”
“Every person with four arms has two heads.”
According to this, I can state with absolute certainty:
“If the two axioms above are true, then I have two heads.”
This statement does not depend on the truth of the ones above. It is true, regardless.
But mathematicians (and logicians) are way more strict than that. They’d say: “you can’t just state “use the laws of logic to infer new truths” and start producing theorems. You need to specify which laws of logic you’re using.” Hence came mathematical logic, which can be seen as the systematization and symbolization of thought. Mathematical logic was taken to its modern form by Russell (sorry, Tongue Sandwich) and Whitehead in their famous Principia Mathematica.
Venn diagram for the set theoretic intersection of A and B. (Photo credit: Wikipedia)
In this new framework, mathematics had just become the manipulation of symbols according to certain rules. The laws of logic (inference rules) determined which new chains could be built from pre-existing ones, starting with a set of arrays of symbols that were just a given (the axioms.) Funnily enough, instead of taking mathematics closer to truth, this abstraction took it further: mathematics, in fact, wasn’t about truth at all. It was just about manipulating chains of abstract symbols, the meaning of which – if they had any – was to be determined later. In fact, determining the meaning (the possible applications) of a certain mathematical theory was not considered part of mathematics, but applied mathematics. Mathematics just dealt with the abstract relationships between symbols, whatever they meant.
The formalization of logic opened the door to alternative logics. If logic is just a set of rules for creating new chains of symbols, can’t we use a different set? The 20th century saw the appearance of many such alternative logics: intuitionist logic, fuzzy logic or quantum logic, to name a few.
This vision of mathematics as the systematic manipulation of symbols opened another door: the possibility of automatization. If all we are really doing is applying pre-determined laws of transformation to a series of symbols, it should be possible to make this process systematic. The arrival of computers (which, by the way, was closely related to the development of mathematical logic) provided such a way. In this sense, a mathematical theory could be seen as a computer program: given the following input (axioms), apply your inference rules (instructions) to find every possible theorem. Mathematics could then be seen as the set of all possible computer programs that can be executed by a machine with infinite memory and capacity (or infinite time to perform its operations.)
After that, things got more complicated. Notions of soundness and completeness were explored and a lot of Earth-shaking results were obtained. I will go into these more deeply in following articles.
For now, I will give your brains a break. I know mine needs one.
PS Question: being a non-native speaker, I’m confused. Mathematics: plural or singular? In Spanish it’s plural. In English I think I’ve seen both uses. Yeah, I know, asking this question about a whole blog on mathematics completely undermines my reliability. Can’t be helped.