My Struggle With Mathematical Philosophy
By Jason Earls, author of Red Zen, Cocoon of Terror, & Heartless Bastard In
Ecstasy http://becomeguitaristfromhell.blogspot.com/ http://www.youtube.com/user/zevi35711
Do numbers exist only in the human mind? Could they be “living” in the Platonic realm somewhere? Do numbers simply exist as symbols on pieces of paper? What is the precise nature and existence of numbers anyway? And does anyone care other than myself? Sure they do. Philosophers have 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 and legitimate theory, despite the many new theories that have popped up in recent times concerning this intriguing age-old problem. Platonism can be defined in this way: mathematics and numbers 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 being discovered any practical purpose for it 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 build up a body of work (by making experiments and noticing patterns, proving and conjecturing, writing papers and books) and over time they construct a body of overall knowledge dubbed ‘mathematics’ that is continually expanded upon, revised, built onto, taught in schools, debated over, etc. Hersch calls this new theory of mathematics, ‘humanism.’ But personally I am not convinced of its correctness, since none of the arguments in his book seem 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, (since the basic laws of the universe still hold over vast distances), the Quorfs will notice that in their counting system, some miglets will not be divisible by any other miglets (primes), while others will have divisors (composites), and then the Quorfs will go on to notice a whole host of rules, laws, and properties for their miglet system. Even if the Quorfs live in a 4-dimensional section of the universe and have eight “digits” on six major appendages and use a base-48 counting system, doesn’t it seem plausible to presume that eventually they will discover the same “laws” and “rules” that our numerical system has? Only that they will call their rules by different names? (The scientists at NASA agree with me; which is why they send representations of primes out in capsules into deep outer space in search of extraterrestrial intelligence – they believe that any life they discover, if it is indeed intelligent, will know what a prime number is.) Hence, even after the human aspect and all 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 known as “number” lives and waits for anything possessing intelligence to discover it. 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 that’s simply the way we humans have done things thus far throughout history; and that other cultures could do things in a different way. But in my opinion, it seems that the concept of “necessity” has dominion here. I.e., practical real-world necessity determines that 8 + 3 = 11, instead of any group’s opinion whether 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 is obviously not a precise integral number. But 365.242 is indeed 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 whatsoever, but they are simply collections of symbols whose overall forms and proofs may have some useful applications, but other than that, they are ultimately meaningless. In other words, formalism says that mathematicians merely manipulate symbols on pieces of paper, following their own arbitrary set of rules, but that it really makes no sense in an ultimate way. Constructivism is another philosophical theory related to mathematics, but it has a much 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 personal 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 would seem 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 highly intrigued with the idea and developed a strong desire to believe in it. But ultimately I could not persuade myself that it was indeed true enough for me to actually leave Platonism behind. In other words, I wanted to believe that constructivism was true since that would make all of my mathematical activity, and everyone else’s mathematical work too, seem “contrived” or “arbitrary,” and that discovering something “beautiful” was not as important or as special as it seemed. But I could not actually make myself believe in constructivism, since my natural intuition kept interfering and telling me that people actually made mathematical discoveries whenever they “did” mathematics. But I still enjoyed the idea of people constructing mathematics then, and I still do today. Here is an example. The mathematical work I engage in now consists mainly of being a “prime hunter,” which means that 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 proving their specific “form;” and then I submit the probable primes I find to a mathematical database. One of my favorite polynomials that has brought forth an abundance of probable primes is the following.
n# * 10^(n+k) +prime(n+j)
It consists of a primorial: #, powers of 10, and the addition of a small prime at the end. With each
new search I regularly change the values of the k’s and j’s and usually it allows me to find more primes. This polynomial has been exceedingly lucrative in terms of plucking primes from the Platonic realm, even though I am not exactly sure why it works so well. I suspect it has something to do with the multiplication of many small primes, and 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 that this polynomial is any more effective than other, simpler, more elegant forms. My main point however is that the polynomial above would appear exceedingly UGLY to a real mathematician! But I still think it’s nice regardless. Why? Because it works! And because it is slightly more complicated than other prime polynomials! But here is my real 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, so inelegant, so thoroughly unattractive. But my intuition tells me that the actual primes it gathers forth are already lurking out there in the Platonic realm and that I am indeed 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 tells me (strongly) that I am a Platonist. Perhaps you would like to pick up a book or two on mathematical philosophy and determine which belief you subscribe to. Have fun and write to me and let me know what you think. Sources: What Is Mathematics, Really?, Reuben Hersh, Oxford University Press, 1999. Wikipedia, Philosophy Of Mathematics, http://en.wikipedia.org/wiki/Philosophy_of_mathematics -end(Thanks for reading. If you have any comments, or know of any magazines that would like to publish this piece, please contact the author: zevi_35711@yahoo.com. Also, you would be helping out the author greatly if you purchased one of his books from Amazon.com or another online book store of your choice. Thanks again.) http://www.youtube.com/user/zevi35711 http://becomeguitaristfromhell.blogspot.com/ http://zombiesofthereddescent.blogspot.com/ Bio: Jason Earls is the author of Cocoon of Terror (Afterbirth Books), Heartless Bastard In Ecstasy, How to Become a Guitar Player from Hell, Zombies of the Red Descent, If(Sid_Vicious == TRUE && Alan_Turing == TRUE) {ERROR_Cyberpunk(); }, Red Zen, and 0.136101521283655... all available at Amazon.com and other online book stores. His fiction and mathematical work have been published in Red Scream, Yankee Pot Roast, M-Brane SF, Scientia Magna, three of Clifford Pickover’s books, Mathworld.com, AlienSkin, Recreational and Educational Computing, Escaping Elsewhere, Neometropolis, Thirteen, Dogmatika, Prime Curios, the Online Encyclopedia of Integer Sequences, OG’s Speculative Fiction, Nocturnal Ooze, Bust Down the Door and Eat All the Chickens and other publications. He currently resides in Oklahoma with his wife, Christine.