00:00:15.360 Is it substantially different from other subjects?
00:00:18.280 Is it just a game or are mathematicians discovering something real about the structure of reality?
00:00:23.400 Welcome to the top cast on the nature of mathematics.
00:00:36.080 The rules of chess are pretty straight forward once you learn them.
00:00:42.280 Yes, the bishop can go as far as you want, but it's restricted to moving diagonally.
00:00:49.160 The rook can also travel from one end of the board to the other, but only down columns
00:00:54.960 The king, any direction at all, but it can only move one square at a time.
00:01:01.640 These few simple rules lead nonetheless to a richer ray of possible games.
00:01:06.760 In fact, the number of possible games, despite being unimaginably large, is in fact
00:01:13.680 One estimate is that it's around 10 to the power of 10 to the power of 50.
00:01:18.080 And aside, the rules of chess seem to be arbitrary, they could have been different.
00:01:23.480 We can imagine that the bishop might have been limited to moving only two squares rather
00:01:31.360 than the length of the board, or we could imagine a game where the queen simply doesn't
00:01:36.160 exist, or that another piece takes the place of half the pawns.
00:01:40.920 The possibilities seem endless, and the game of chess we do have is simply the result
00:01:57.880 Is it just a self-consistent set of rules that permit games to be played?
00:02:02.920 Could we change the rules, and so play a different game?
00:02:09.600 It is a popular idea, and it permeates much of academia, especially among some in the humanities.
00:02:16.840 This is not to say that all and only those in the humanities think this way.
00:02:21.160 Certainly some mathematicians believe they are simply playing games.
00:02:24.720 Famously, David Hillbert, the great German mathematician, who worked on the very foundations
00:02:30.440 of proof theory, probably thought something like a version of this idea.
00:02:35.360 But he's notable as being an exception in this regard among mathematicians, and anyway,
00:02:40.200 funny as it may seem, a mathematician is, in fact, not necessarily the best person to consult
00:02:47.720 As we will come to see more and more in the theory of knowledge course, while we should
00:02:52.160 never rely solely on any authority, a philosopher, in this respect, is likely to have
00:02:57.960 a more informed and considered opinion, in particular, in this case, a philosopher of
00:03:04.880 This is kind of similar to the way that a pilot of a 747 might be a highly competent
00:03:10.280 user of an aircraft, and yet not have a very complete knowledge about how precisely the
00:03:15.040 wings and engines come together to allow him to do his job.
00:03:18.640 Sure, he might have a very good idea, but really, it's not essential to his day-to-day
00:03:23.280 business to know that much about aeronautical engineering, an aeronautical engineer
00:03:39.360 Philosophers of mathematics in general do seem to be realist about the nature of mathematics.
00:03:46.360 Mathematicians themselves are also generally realist about their own subject.
00:03:52.040 Realism means that there is a reality out there independent of the human mind.
00:03:57.360 Radical realism, therefore, is the idea that mathematical entities are also independent
00:04:04.240 This means mathematics is discovered, not invented.
00:04:08.920 Relativist philosophers, on the other hand, generally all have this in common.
00:04:21.240 This may cause one to wonder about the truth of relativism then, is relativism true.
00:04:27.720 If it's true, then there is at least one truth that is not a mere matter of perspective,
00:04:34.600 But then why should relativism be privileged except on dogmatic grounds?
00:04:38.640 I'll have more to say about relativism and the related theory of postmodernism in a future
00:04:45.800 As all such relativist ideas on the nature of reality can be similarly seen to collapse
00:04:51.160 when the foundational self-contradiction that props them up begins to crumble, so too can
00:04:55.840 the relativist view of mathematics be criticized.
00:05:07.800 Mathematical truth is not a mere matter of taste.
00:05:11.080 Unlike chess, the rules cannot be so easily changed.
00:05:15.520 When you wished to change the theorem that 1 plus 1 equals 2 to become 1 plus 1 equals
00:05:23.760 Is it merely by convention we say one of these is true and one is not?
00:05:28.480 Well, imagine we did suddenly decide that 1 plus 1 equals 3.
00:05:34.200 Well, the word 3, and the symbol for it, would come to mean what we mean now by 2.
00:05:40.800 Then what happens to the abstract entity to which 3 previously referred?
00:05:45.400 Well then, we would need a new word and a new symbol, perhaps we could use 2.
00:05:52.520 Whatever we did decide to use, the entire edifice of the remainder of our mathematics
00:05:56.920 would change and it would change in order to consistently accommodate these changes.
00:06:02.360 And in the end, we would have changed nothing but the symbols.
00:06:07.320 It is worthwhile pointing out here that mathematics works.
00:06:11.920 Mathematics is not merely an invention of the human mind.
00:06:14.680 Let's say the great trilogy, Lord of the Rings, because mathematics enables us to make
00:06:21.320 If mathematics were a mere invention, then we should not expect it to be so accurate in
00:06:30.080 No work of fiction is able to predict, with anything approaching the same level of fidelity,
00:06:35.000 the things that mathematics, in partnership with science, can.
00:06:39.360 The laws of physics are expressed mathematically, and the fact that they are mathematically
00:06:43.160 informed is not only somewhat mysterious, but also fortuitous.
00:06:48.440 Imagine that we wish to calculate exactly how long it will take for a rock dropped from
00:06:56.600 Physics tells us that if we know just a few things, like the height of the cliff and the
00:07:01.960 force of air resistance, then we can write down a mathematical formula, perform a calculation
00:07:07.680 and make a prediction with an extremely high degree of accuracy.
00:07:12.560 If mathematics were simply all made up, and was just a game, why should we expect it to
00:07:19.920 It seems that mathematics is not something we can make up any more than any other part
00:07:25.960 Mathematics is there to be discovered, and we can be truly right or wrong about it.
00:07:32.600 When we look at something purely mathematical, like say the distribution of prime numbers,
00:07:40.360 The prime numbers are distributed in a way we do not completely understand.
00:07:44.560 We cannot predict exactly when, among all the natural numbers, the next prime number will
00:07:50.520 The highest prime number found, as at the end of 2009, has around 13 million digits.
00:08:02.760 Euclid, in 300 BC, was able to prove that there are infinitely many primes.
00:08:16.120 Well, it does not yet exist in any physical way.
00:08:19.600 And so this is why we say mathematics is about abstract entities.
00:08:24.040 As far as prime numbers go, however, it has even been suggested that communication with
00:08:28.200 alien intelligence could, in part, be recognized by an embedding of prime numbers in a radio
00:08:38.400 In other words, we should expect aliens who would speak none of our terrestrial languages
00:08:43.520 would nonetheless know the same basic mathematics, especially if they are capable of using
00:08:48.720 electromagnetic radiation for communication purposes.
00:08:52.360 It turns out it would not matter if the number system they used was based 10 like ours,
00:08:56.880 based 60 like the Babylonians, or based two like a computer, or any other base.
00:09:01.600 The prime numbers are somehow out there to be discovered.
00:09:05.160 And you can express the same number using any system you like.
00:09:09.360 The prime number that in base 10 we know as 7 is 0, 1, 1, 1 in binary.
00:09:18.600 They are a series of numbers, some of which have been discovered, and an infinity more
00:09:25.480 Again, we come back to the point that mathematics in general is something that is discovered,
00:09:32.320 There are many flavours of realism, which is the alternative to the relativist position
00:09:37.920 It is worth looking into, for example, Platonism, Logicism, and formalism in more detail.
00:09:44.720 But the type of realism I will discuss is a type of what is called empiricism.
00:09:49.480 But for the sake of clarity, I'm simply going to call it the realist perspective.
00:09:54.360 The idea is this, mathematics is about necessary truth.
00:10:03.120 But in the end, pure mathematics is the study of necessary truth.
00:10:08.840 It is important to distinguish between what are known in philosophy as contingent truths
00:10:27.560 Well, a necessary truth is one that could not have been otherwise.
00:10:32.720 It is true by virtue of the meanings of the terms that make it up.
00:10:36.600 For example, a triangle has three sides is a necessary truth.
00:10:41.200 The meanings of the words triangle, three, and sides in this sentence mutually define
00:10:48.280 I don't even need to draw the triangle for you to know the statement is true.
00:10:52.920 If I drew something that was foresighted and called it a triangle, I would be wrong.
00:10:57.800 This reminds me of a joke, a philosophy lecturer of mine once told me.
00:11:01.840 If you call a horse's tail a leg, how many legs would it have?
00:11:20.440 You can call the tail whatever you want, a horse has four legs.
00:11:27.200 It is simply not possible for them to be otherwise without contradiction.
00:11:32.840 If you are in Sydney, then you are not in Melbourne.
00:11:41.120 It is necessarily the case that one plus one equals two.
00:11:44.760 It cannot be otherwise, so you can't simply change the rules.
00:11:48.760 They can actually rigorously prove, for example, that the theorem one plus one equals two.
00:11:54.120 It actually takes around half a page using something called penis axioms of arithmetic.
00:12:02.760 These are things like that for any number x, then x plus zero equals x.
00:12:08.120 And for any number x, if you add one, then you get the immediate successor of x and so
00:12:13.800 I'll be coming back to what axiomatic systems are all about in the next talkcast.
00:12:25.160 And so which entail a contradiction if you deny them?
00:12:29.040 What about that other sort of truth, I mentioned?
00:12:34.480 contingent truth is the truth of all natural and human sciences.
00:12:41.520 For example, the true statement, the earth is the third planet from the sun, is a contingent
00:12:47.440 You can certainly imagine a possible world where Venus did not exist, making earth the
00:12:59.680 You might be wrong, but there is no contradiction in saying World War II ended in 1994.
00:13:05.240 There is nothing about the definitions of World War II and ended in 1994 that entire
00:13:13.480 There might be some empirical facts that lead to the statement being false, but it is
00:13:18.280 false based upon knowledge of the physical world.
00:13:21.560 If one never knew anything else about World War II, other than it began in 1939 and
00:13:26.240 ended in 1994, one might go through life with this falsehood.
00:13:32.680 On the other hand, one cannot consistently believe in foresighted triangles.
00:13:38.400 So there is a distinction to be made here between what is a necessary truth and what is
00:13:45.480 This distinction, it turns out, is also important in considerations about the existence
00:13:52.200 Leibniz, one of the inventors of calculus, along with Newton, was one of the greatest philosophers
00:13:57.280 of all time, and was able to solve the problem of free will in a universe with an omniscient
00:14:02.520 god that is a god who knows everything, including all of your future actions.
00:14:07.040 Whether you believe in an omniscient god, or whether you believe that the laws of physics
00:14:10.560 ultimately determine all of your future actions, Leibniz is able to reconcile these views
00:14:15.560 with free will, and that is something I'll talk about in a future talkcast.
00:14:28.400 Now just because mathematics is about necessary truth, does this make it certain?
00:14:34.360 It turns out we cannot be certain about mathematical truths.
00:14:37.760 This is because in order to establish something as mathematical fact, we need to prove it.
00:14:43.880 Now a proof, a mathematical proof, or a purely logical proof, is a process.
00:14:50.240 It is a process that usually a person goes through using pen and paper, or a calculator,
00:14:58.720 I say usually, because some proofs today are completed entirely by computers.
00:15:14.280 Whether the theorem one is interested in, is written in symbols on paper, or it is simply
00:15:18.840 conceived of in the mind, or it appears on a computer screen, or it is calculated using
00:15:23.880 a machine, the process of proof relies upon some physical system.
00:15:34.000 Mistakes can be made on paper, and brains can make errors.
00:15:38.040 All of this adds up to say that mathematical theorems can only ever be as certain as the
00:15:43.080 proofs used to establish them, and because a proof is necessarily a physical process, it
00:15:49.080 is going to be subject to the same fallibility as our knowledge about any physical process.
00:15:55.800 In short, proofs are not abstractions, they are computations, and a computation always
00:16:02.040 requires a computer, and it does not matter if that computer is made of silicon or biological
00:16:10.520 This point of view is not shared by all mathematicians and philosophers.
00:16:15.440 It is worth looking at alternative points of view, including Platonism.
00:16:20.760 Platonism is in the idea that mathematical entities exist in another abstract realm that
00:16:28.920 Importantly, the proof process on this platonic view is generally also regarded as an abstract
00:16:35.400 process that can be separated from the physical world, and therefore we can get mathematical
00:16:40.920 certainty by somehow tapping into the non-material abstract world of platonic forms using
00:16:49.960 Some other philosophers think this view is misguided, and the confusion stems in part from
00:16:54.560 confusing the processes of mathematics with its subject matter.
00:16:59.200 Mathematics is about necessary truth, the truths that mathematics studies are themselves
00:17:04.040 certain, but this does not mean our knowledge about these truths is certain.
00:17:09.080 As David Deutsch so succinctly puts it, necessary truth is merely the subject matter
00:17:13.880 of mathematics, not the reward we get for doing it.
00:17:17.760 The objective of mathematics is, therefore, not mathematical certainty, or even mathematical
00:17:27.720 Even in proofs which do not involve diagrams, we must still use symbols or calculators
00:17:35.440 All of these things are physical objects, and so the argument still holds.
00:17:40.200 That is, our knowledge of mathematics is based upon our physical theories about how physical
00:17:47.000 For this reason, we can be no more certain about the truths of mathematics than we can
00:17:54.920 Ultimately, our confidence about how physical objects behave is based upon our scientific
00:18:00.760 theories about how those physical objects behave.
00:18:05.120 Finding the relationship between those physical objects and the mathematical proof we are
00:18:09.000 undertaking with them, be that pen and paper, a computer or a brain, tells us how confident
00:18:14.880 we are entitled to be about our deductive proofs.
00:18:26.480 It is sometimes assumed, at least tacitly, that there is a hierarchy of argument in academic
00:18:33.160 There are mathematical proofs that are absolutely certain.
00:18:36.920 There are scientific arguments that, while not absolutely certain, are certainly still
00:18:42.600 And finally, there are philosophical arguments, and there are mere matter of taste.
00:18:47.280 Here, I have used a philosophical argument to establish that mathematical explanations
00:18:54.320 rely upon our scientific theories for their veracity.
00:18:58.680 It is simply not true that mathematics gets us certain truth.
00:19:02.720 It is useful, it can be beautiful and elegant, and it is essential in science, but it
00:19:10.200 Really, all claims about the nature of reality and truth are philosophical.
00:19:15.520 Philosophical arguments should be logical and so consistent, and based on as few assumptions
00:19:22.120 In the end, it is a good philosophical argument that gets us to our understanding of mathematics
00:19:27.120 as a subject about necessary theorems, whose truths are established by the physical
00:19:39.600 Or could we one day program a computer, who will be able to prove all of those necessary
00:19:46.000 Well, that is the subject of the next talkcast.
00:19:57.960 Our resources and links associated with talkcast visit the talkcast website at www.talkcast.net.
