Philosophy of Numbers in Math
Do numbers exist only in the human mind? Could they be “living” in a Platonic realm somewhere? Do numbers simply exist as symbols on pieces of paper? What is the precise existence of numbers anyway?
Philosophers been arguing over the true nature of mathematics and numbers for thousands of years. But in my view, Platonism still holds up as the most truthful theory, despite the many new ones that have popped up in recent times concerning this age-old problem. Platonism can be defined as… numbers and mathematics are abstract concepts that have an independent existence apart from the symbols or terminology used to represent them; i.e. the number Pi = 3.141592653… existed somewhere in the “Platonic realm” long before any human ever discovered its practical purpose here on Earth.
In the book, What Is Mathematics, Really? By Reuben Hersh (Oxford University Press, 1999) Hersh puts forth the proposition that mathematics exists simply as a social activity performed by a certain class of humans who gradually work (make experiments and notice patterns, prove and conjecture, write papers and books) to gradually build up a body of knowledge called ‘mathematics’ that is continually added to, debated over, expanded upon, etc. Hersch calls his new theory of mathematics, ‘humanism.’ But personally I am not convinced, as none of the arguments in the book seemed strong enough to counter the following argument.
Say a new form of life arises in the future somewhere in the Black Eye Galaxy (NGC 4826; which was discovered in 1779 by Edward Pigott) and the “aliens” there call themselves “Quorfs.” Say these quorfs gradually develop a system of counting, but instead of numbers they call them “miglets.” Time goes by and gradually the quorfs develop rules for their miglet system and eventually notice rules and laws governing their behavior. Surely at some point the Quorfs will notice that some miglets are not divisible by any other miglets (primes), while others do have divisors (composites) and then they will notice a whole host of rules, laws, and properties for their miglet system.
Even if the Quorfs live in a 5-dimensional world and have eight “digits” on six major appendages and use a base-24 counting system, doesn’t it seem plausible that eventually they will discover the same “laws” and “rules” that our number system has? Only they will call things by different names? (The scientists at NASA believe this is true, which is why they send out primes in capsules to deep outer space in search of extraterrestrial intelligence.) Hence, even after the human aspect and any earthly subjectivity has been removed, there will remain some objectivity pointing the way to a realm of forms living in an independent reality, where the concept of numbers lives and waits for anything possessing intelligence to discover.
Ludwig Wittgenstein, one of the greatest philosophers ever to take a breath, said that the laws of mathematics are not self-enforcing, so if people agreed to do mathematics a completely different way than our present method, they too would be correct in whatever rules they happened to formulate. For example, Wittgenstein said that 8 + 3 is not truly equal to 11 in absolute reality, but that it’s just the way we humans have done things so far; and that other cultures could do it differently. But to me it seems that “necessity” has dominion here. That is, practical, real-world necessity determines whether 8 + 3 = 11, instead of any group’s opinion that 8 + 3 equals 11.
Do we humans apply numbers to the universe arbitrarily just so we can make sense of it? There are slightly more than 365 days in a year, about 365.242, which obviously is not a precise integral number. But 365.242 is still a number, just as Pi = 3.141592653589… is a number that’s irrational and transcendental, yet highly useful.
There is also a debate in the world of mathematical philosophy concerning whether numbers are discovered or invented. Formalism is a theory in philosophy that says mathematical statements actually have no meaning at all, but they are simply collections of symbols whose overall forms and proofs may have some useful applications, but other than that they are meaningless. In other words, formalism says that mathematicians merely manipulate symbols on pieces of paper, following their own arbitrary rules, but that it really makes no sense in an ultimate way.
Constructivism is another philosophical theory related to mathematics, but is has a more complicated definition, which I will have to quote: “… in the philosophy of mathematics, [constructivism is] a broad position (encompassing both intuitionism and formalism but also going beyond them) which holds that mathematical entities exist only if they can be constructed and that proof and truth in mathematics are co-extensive. Constructivists oppose the realist (or Platonist) view that mathematical objects or truth exist independently of human procedures. This has the consequence that certain classical results whose proof rely on Platonic assumptions are not constructively valid.” – From The Blackwell Companion to Philosophy.
The definitions above bring me to a story which may help elucidate things further: I used to be addicted to submitting integer sequences to the Online Encyclopedia of Integer Sequences. Occasionally I would construct a sequence that was highly artificial. At those times it seemed that I was “inventing” mathematical objects instead of “discovering” them. I had plenty of sequences rejected from the OEIS because of my arbitrary constructions. Around this time I also learned of the concept of constructivism and became intrigued with the idea and wanted to believe in it. But I couldn’t persuade myself it was true enough to actually leave Platonism behind.
In other words, I wanted to believe that constructivism was true since that would make all mathematical activity “contrived” or “arbitrary,” and that discovering something “beautiful” wasn’t as important or as special as it seemed. But I couldn’t actually make myself believe in constructivism since my natural intuition kept telling me that people made mathematical discoveries whenever they “did” mathematics. But I still liked the idea of people constructing mathematics, and I still do.
Here is an example. The mathematical work I engage in now consists mainly of being a “prime hunter,” which means I search for large ‘probable primes’ – those that pass a Fermat test but that cannot officially be certified prime since no one knows an elegant algorithm for doing it, and then I submit the probable primes I find to a math database. One of my favorite polynomials that has brought forth an abundance of probable primes is this:
n# * 10^(n+k) +prime(n+j)
Which consists of a primorial: #, powers of 10, and the addition of a small prime on the end. With each new search I regularly change the values of the k’s and j’s and usually find more primes. This polynomial has been exceedingly lucrative in terms of plucking primes out of the Platonic realm, even though I’m not sure why it works so well. I suspect it has something to do the multiplication of many small primes, the abundance of 2’s and 5’s from the powers of 10, and finally the addition of a prime on the end which increases the chances of finding another larger prime, etc. But I cannot prove this polynomial any more effective than another simpler more elegant form.
My main point however is that the polynomial above would appear quite ugly to a real mathematician! But I still think it’s nice. Why? Because it works and it’s slightly more complicated than other prime polynomials! Now here’s the main point: Did I create this form of mathematics? Or did I discover it? It feels like I created it since the polynomial seems so artificial, inelegant, and unattractive. But my intuition says that the actual primes it gathers are already lurking out there and that I’m discovering them. There is no way to prove it one way or another.
So, despite much philosophical meandering, the Platonic Realm in my view is where numbers still seem to exist. And even though I am intrigued by the idea of constructivism, my intuition still says that I’m a Platonist. Perhaps you would like to pick up a book or two on mathematical philosophy and determine which belief you subscribe to.