00:00:00.000 Mathematics. Why? For a right-angled triangle. Is the square of the hypotenuse equal to the sum of the squares of the other two sides?
00:00:08.880 Why? Almost everyone learns that c squared equals a squared plus b squared.
00:00:15.920 Asking why is asking for the explanation? In mathematics,
00:00:21.120 explanations often involve a particular structure or process, and that process is known as proof.
00:00:27.120 If there are many proofs of the Pythagorean theorem, some are algebraic and begin with axioms, others are geometric.
00:00:34.400 Some graphical or even animated. A book exists containing
00:00:38.480 367 different proofs of the theorem. In all cases, the proof constitutes the mathematical explanation.
00:00:46.400 Proofs exist, so one can be shown, which is to say, understand why c squared equals a squared plus b squared for a right-angled triangle.
00:00:55.200 But proofs start somewhere. They begin with axioms, axioms are things not proved.
00:01:01.120 They are what is assumed true, and indeed are sometimes simply called the assumptions.
00:01:09.120 Many arguments are put forth here, often coming down to, it is self-evident.
00:01:13.600 It couldn't be otherwise, a falloblast might say.
00:01:16.400 Well, we do not know they are true. We're just assumed, tentatively, they are.
00:01:21.840 Or perhaps we just say, they are the gnomes. They are what we know now,
00:01:26.800 and we also know the rules of inference, and together we can see where they lead to what conclusions or
00:01:32.240 theorems. In a worldview that values explanations, we might even say, the axioms are what they are
00:01:38.320 in mathematics, because no explanation is ever improved by changing any of them.
00:01:43.280 Those axioms themselves constitute explanations of a kind, claims about reality.
00:01:49.760 Not certainly true, but representations of necessary truth.
00:01:54.400 We are all of us, fallible, even mathematicians.
00:01:57.840 Mathematics commands a special place in academia.
00:02:00.880 Traditionally, it is the subject to which all others have aspired.
00:02:04.880 Mathematics, it has been thought, are where we finally get to so-called a epistemological bedrock.
00:02:11.120 Some certain and absolute truth where all doubt is removed,
00:02:15.200 science can only aspire to such lofty heights, the science relies on evidence.
00:02:20.000 Open to interpretation, and so it is said, only the accumulation of sufficient amounts of it
00:02:25.680 can ever make us confident our theories are true, or probably true, as for philosophy and
00:02:31.840 everything else. Me, matters of opinion, of course that entire hierarchy of mathematics as certain
00:02:38.640 science is probably true, and everything else is more or less a matter of opinion is itself
00:02:44.000 not a mathematical claim. It's a philosophical one.
00:02:47.600 Instead, mathematics is better understood as distinguished by its subject matter and method.
00:02:53.920 That method of mathematical proof does not mean show as true what it means is,
00:02:59.360 calculate or compute what follows given some axioms using some rules of inference.
00:03:05.680 Mathematics is about explaining necessary abstract truth,
00:03:09.680 but as David Deutsch wrote in the fabric of reality in his chapter on mathematics, quote,
00:03:14.560 necessary truth is merely the subject matter of mathematics. It is not the reward we get for
00:03:20.400 doing mathematics, end quote. In a sense, much of what David Deutsch's famous for itself
00:03:27.040 proves that very conclusion, in laying the foundations for quantum computation,
00:03:32.240 David proved what was then sometimes called Turing's thesis, which may be expressed as
00:03:37.600 every physical process is computable. That implies anything in physical reality can be
00:03:43.360 simulated by a computer from atoms to weather systems, galaxies even in principle a human brain.
00:03:50.640 David Deutsch took the thesis and on the assumption of quantum theory proved it.
00:03:55.280 After all, computers and brains are not abstract objects. They are made of matter,
00:04:00.560 matter obeying laws of physics, laws of quantum physics. So now many call it the
00:04:05.920 church Turing Deutsch principle to distinguish it from a mere thesis of pure mathematics.
00:04:11.680 It is now properly regarded as part of physics, a principle governing the behaviour of all
00:04:17.600 physical objects, and that includes brains like the brains of mathematicians.
00:04:22.480 The laws of quantum physics say we can never rule out the possibility of something unexpected
00:04:28.720 happening. This is otherwise explained as there is always subjective randomness. In other words,
00:04:34.000 you cannot know the outcome exactly of any physical process there will always be uncertainty,
00:04:40.000 a range. This applies to brains, classical and quantum computers, any process. It applies to
00:04:45.520 what mathematicians do. Content theory mandates we can never know with absolute mathematical certainty
00:04:52.000 that a particular outcome must happen. There is always a range. However, small the chance of error,
00:04:57.920 it does not matter, it is not zero is the point. So physical systems evolve in ways we
00:05:03.600 cannot know for sure. And that includes brains and the outcomes of any computation.
00:05:11.040 And mathematical proof constitutes a computation and therefore cannot confer upon its conclusions
00:05:17.440 absolute certainty. Not only because we cannot know for certain the axioms are true but because
00:05:23.280 the process itself of mathematical proof is a computation. Computations are done by computers
00:05:28.960 and computers obey the laws of physics that rule out, knowing for certain that something unexpected
00:05:35.360 did not just happen during that proof. Mathematics is the study of necessary truth.
00:05:40.640 It constructs knowledge of necessarily perfect abstractions but the knowledge of that perfection
00:05:46.880 is not the perfection itself. Any more than knowledge of electrons is electrons or knowledge of birds
00:05:52.480 is birds, we are fallible and knowledge is imperfect even in mathematics.
