00:00:00.000 Are there truths out there that will forever remain unknowable?
00:00:19.760 Can mathematics have any input into this question one way or the other?
00:00:23.520 In this topcast, we're going to look at the limitations of mathematics and some more of
00:00:28.160 the curious and surprising consequences of deductive systems.
00:00:36.000 Perhaps you're having a drink while listening to this.
00:00:38.560 I've got a cup of tea here with me at the moment, it's about a metre away from my hand.
00:00:44.000 In order to pick up my tea for the next sip, I can think of my hand moving in steps like
00:00:53.920 Once that's done, I move my hand a quarter of a metre towards the cup.
00:00:58.400 After that, an eighth, then a sixteenth, and so on.
00:01:03.120 In other words, I can divide up the metre between my hand and the cup into an infinite
00:01:08.520 number of ever smaller increments, half of the distance, then half of what's left,
00:01:16.720 I would have an infinite series, a half, a quarter, an eighth, a sixteenth, a thirty-second
00:01:26.640 But if this sequence is infinite, even if the increments get ever smaller, it will take
00:01:32.200 me an infinite amount of time to reach the cup, because each step, however small, will
00:01:42.160 And I don't have an infinite amount of time, yet I can pick up the cup.
00:01:53.320 It's obvious I can pick up the cup, and yet I've produced an argument that seems to
00:02:11.680 As we live in a physical world, the physical argument is the true solution in the actual
00:02:19.920 According to quantum physics, you simply cannot divide up either space or time into infinitely
00:02:27.760 There is actually a limit on how small space can be, and how short time can be.
00:02:33.520 The division of time will go on for a while, but eventually it will terminate at a shortest
00:02:39.640 In fact, it stops at around ten to the power of minus thirty-four seconds.
00:02:49.480 There is, however, also a purely mathematical solution as well.
00:02:53.960 It says, it doesn't matter if you have a series made up of an infinite number of terms.
00:03:00.000 You can add up all those terms, and yet still get something less than infinity.
00:03:05.040 If you do a higher level of mathematics, you might like to prove what the series, a half,
00:03:10.240 plus a quarter, plus an eighth, plus a six-ninth, onto infinity, adds up to.
00:03:15.320 Or more rigorously, what the limiting sum of that series is.
00:03:22.880 In other words, I can add up all those little fractions and still get a finite number.
00:03:28.200 Remember, I started with something finite, a meter, and then basically argued for cutting
00:03:36.840 I then tried to say that you couldn't take all of those slices and add them together again
00:03:47.000 This is essentially one of the paradoxes of Zeno.
00:03:50.400 Zeno of Elia came up with these in ancient Greece and around 450 BC.
00:03:55.960 Another of these arguments was about the tortoise and Achilles, and that's worth looking
00:04:03.360 It suggests that deductive systems can sometimes turn up some really surprising results.
00:04:10.200 In the case of the paradox just discussed, it seems like I had a watertight argument,
00:04:17.560 Of course, it turns out my argument was not a real paradox.
00:04:21.840 But can you have real paradoxes in mathematics?
00:04:26.480 We will see that paradoxical statements actually have consequences for the entirety of
00:04:35.800 The students of mathematics at some point encounter set theory in their senior years
00:04:47.360 In fact, even primary school students can be taught the basics of set theory using Venn diagrams.
00:04:52.960 This sort of set theory allows for everything in the universe to be considered part of
00:05:00.160 We can talk about infinite sets or finite sets.
00:05:03.760 The set of all positive integers is an infinite set.
00:05:07.040 The set of all prime ministers of Australia is a finite set which has 26 members as of
00:05:18.560 Now does this idea of sets lead to any problems?
00:05:24.600 Bertrand Russell, a British mathematician, who was alive in the early to mid-20th century
00:05:29.880 and who will hear about more later on, raised a version of the following paradox.
00:05:34.720 It is named for him and so it's called Russell's Paradox.
00:05:38.120 The version I will give you is not exactly like Russell's own, but it's the one which
00:05:41.800 appears in most texts and resources on the internet.
00:05:55.920 A catalog is a list of all books in the library.
00:05:59.360 Now imagine that the catalog is a physical book, listing all the books in a particular
00:06:05.480 Now most libraries have the practice of including a reference to the catalog in the
00:06:11.000 After all, the catalog itself is a book and a library, so we can written down in the
00:06:15.480 But I happen to know that my local library does not list the catalog itself as an entry
00:06:21.360 The library and thinks it would be silly to do so.
00:06:24.000 After all, if you've got the catalog in your hands, you don't need to look it up to find
00:06:31.120 The most important library in Australia is called the National Library, and it's in Canberra.
00:06:37.040 Now the National Library keeps a record of all the library catalogs in the country.
00:06:45.080 The first contains all the catalogs which list themselves as members.
00:06:50.280 The second contains all the catalogs which do not list themselves as members.
00:06:54.960 The problem for the National Library is, where does the entry for the second catalog go?
00:07:00.320 Should we write its entry into the first catalog?
00:07:03.160 If we do this, then we are saying that it lists itself as a member.
00:07:07.280 But that means we need to actually write down an entry for the second catalog into the
00:07:12.520 But once we do this, then we have an entry in the second catalog that breaks the rules
00:07:16.400 for the second catalog, namely that it should list only those catalogs that do not list
00:07:26.400 It's hard work and you may need to listen back over the last couple of minutes to appreciate
00:07:31.120 There are notes to accompany this podcast, and they might help also.
00:07:35.000 I see this particular paradox as a more intricate version of the Liar paradox.
00:07:40.680 The most trivial Liar paradox is, this statement is a lie, or this statement is false.
00:07:47.880 Now consider that statement, is it true or false?
00:07:53.520 If it's asserting something true, then it's a lie.
00:08:02.400 There are many versions of this paradox, and Russell's paradox also.
00:08:06.000 The first such paradox that appears in Western philosophical writings is Epimenities Paradox.
00:08:11.720 Epimenities was from Crete, and he wrote that, all cretins are liars.
00:08:15.800 It's not clear if he intended this statement to be paradoxical or not.
00:08:20.160 Russell's paradox about sets is more succinctly expressed as, if S is the set of all sets
00:08:25.520 that are not members of themselves, is S, a member of S. Again, this is very tricky to
00:08:31.400 get the first time you hear it, and for that reason we went to the National Library in
00:08:37.760 To overcome these problems with set theory, mathematicians choose a set of rules to govern
00:08:44.040 These rules are the foundation upon which set theory is built, and explain what can and what
00:08:49.000 cannot be regarded as a set, and how it is possible to, for example, add two sets together
00:09:03.480 They certainly shouldn't permit falsehoods or absurdity to be proved.
00:09:07.520 It turns out you can define set theory in such a way as to avoid Russell's paradox and
00:09:13.920 Step theory that permits paradoxes is called naive set theory, for this reason.
00:09:22.040 Just pick a set of rules which make paradoxes impossible?
00:09:25.280 Then can we say mathematics will let us prove things as true?
00:09:29.480 The problem after all is that a paradox shows that there is something lacking in your
00:09:35.680 If it permits paradoxes, then it seems like your foundations lead to nonsense.
00:09:41.160 And this might mean your foundations are therefore nonsense themselves, and so the rest
00:09:45.280 of what you end up proving might not be as certain as you hoped.
00:09:49.360 So a proper choice of axioms can eliminate Russell's paradox.
00:09:52.680 Well, yes, but it turns out there are bigger problems than this.
00:10:05.200 At the turn of the century, Bertrand Russell, whom I mentioned earlier, along with Alfred
00:10:09.880 Whitehead, wrote a huge book called Principia Mathematica.
00:10:15.000 In it, they set out to put all of mathematics on a firm logical footing.
00:10:19.360 It was published in 1913, and it was all about the foundations of mathematics.
00:10:24.840 It included the axioms of arithmetic, for example.
00:10:27.680 It included axioms for other types of mathematics also, such as set theory.
00:10:31.640 The question was, could you ever find a statement written down using the rules of mathematics
00:10:40.320 Well in 1931, the answer was found, and it was found by one of the most brilliant mathematicians
00:10:49.440 Kurt Gertel was a German mathematician who was born in 1906 in what is now the Czech Republic.
00:10:55.680 By age 18, he had mastered university-level mathematics.
00:10:59.600 In 1931, he published a paper which contained what are now known as Gertel's infinpleteness
00:11:05.920 They are possibly the most important theorems of mathematics discovered in the last 200
00:11:11.280 Gertel had to actually invent a new form of mathematical proof involving what is now known
00:11:15.000 as Gertel numbering in order to establish his theorems.
00:11:19.240 The proofs themselves are very deep and very complex and run to around 20 pages.
00:11:24.040 You can actually buy them through amazon.com and a little book.
00:11:27.680 It seems like every second popular book on science and every popular book on mathematics
00:11:37.240 Firstly, the concepts of soundness and completeness.
00:11:41.920 Soundness is the concept that a set of rules, logical or mathematical, permit only true
00:11:48.600 In other words, starting from the axioms, you cannot prove a falsehood.
00:11:53.480 Completeness is effectively the mirror image of soundness.
00:11:57.240 Completeness is about whether or not everything that is true within the system has a proof,
00:12:01.680 or equivalently, whether for any given expression you can write down using the symbols
00:12:06.080 of that system, whether that statement can be shown to be either true or false.
00:12:13.080 The most basic useful one is probably centencial calculus or centencial logic.
00:12:20.400 At school, if you're lucky, you might encounter a rarefied version of it.
00:12:24.320 At university, if you take on a science or engineering course, it's hard to miss it.
00:12:27.960 If you study philosophy on mathematics and e-depth, you will likely take on one or more
00:12:34.360 It can pretty easily be shown to be sound and complete.
00:12:38.880 But for systems more complex than baby logic like this, it gets more interesting.
00:12:44.160 Until girdle, no one was really sure about normal run-of-the-mill primary school counting
00:12:50.560 Now the rules or axioms of arithmetic are quite complex.
00:12:55.000 The ones that are generally used, and which girdle used, are called penis axioms after
00:12:59.680 the mathematician who first wrote them down, I mentioned them in the last podcast.
00:13:07.800 Everything that you can prove using the axioms is true, and you can't prove anything
00:13:16.680 Girdle showed that he could write down a statement that asserted its own impossibility
00:13:23.440 Again, to explain this fully takes the full arsenal of Girdle's proof and is well beyond
00:13:28.120 most, except perhaps those who have passed through a third-year university course on the
00:13:36.560 Girdle essentially wrote down the mathematical equivalent of, this statement is false,
00:13:47.240 It suffers from all the paradoxical issues that we discussed early with sets that are
00:13:51.280 not members of themselves, and the liar paradox.
00:13:56.520 Well, that means that mathematics, even simple arithmetic, is not complete.
00:14:02.800 There are statements in mathematics that are true, but which we can never prove as such.
00:14:09.280 And there are statements which we can never decide as being either true or false.
00:14:15.040 It has consequences for computer science, for example.
00:14:18.400 No computer can be programmed, which is able to once and for all solve all the problems
00:14:28.960 It means mathematics will forever remain a creative exercise.
00:14:33.880 Girdle himself was a troubled man, but he became friends with Einstein when he moved to Princeton
00:14:38.160 University in the United States prior to the Second World War.
00:14:41.560 Apparently, Girdle and Einstein would take long walks together at the Institute for Advanced
00:14:45.360 Study where they both worked with Einstein actually remarking that towards the end of his
00:14:49.720 His own work there ceased to interest him very much, and instead he only went in order
00:14:53.600 to have the privilege of walking home with Girdle.
00:14:56.440 For Einstein's 70th birthday, Girdle's present to him was a set of paradoxical solutions
00:15:01.160 to general relativity, which suggested the possibility of time traveled to the past.
00:15:06.440 Towards the end of his own life, Girdle was plagued with mental illness, and became convinced
00:15:13.520 He eventually starved himself to death and died in 1978.
00:15:17.520 I did hear a story years ago that during their walks, Einstein would force Girdle to
00:15:22.280 eat morning tea with him, when he noticed how thin Girdle was.
00:15:26.040 But I can't find any references now to this particular story, and would appreciate it
00:15:35.320 The significance of Girdle's incompleteness Theorems are far reaching.
00:15:39.600 But like quantum theory and cosmology, and any sufficiently complex human intellectual
00:15:43.560 endeavor, it is being commandeered by some to push fringe or even pseudo-scientific claims.
00:15:49.800 Often Girdle's proof is cited by relativist philosophers to bolster their claim that mathematics
00:15:57.240 Without fully appreciating the significance of the theorem or understanding entirely the concept
00:16:01.680 of completeness, you will nonetheless encounter pronouncements that Girdle's proof shows
00:16:12.640 It is also sometimes asserted that the incompleteness Theorems have far-ranging consequences
00:16:22.160 Girdle's incompleteness Theorem leads to the idea that if you have a system like a computer
00:16:26.440 for proving Theorems, let's say of arithmetic, then there will be statements that can
00:16:30.880 be produced using the rules of that system which are true, but which nonetheless cannot
00:16:38.840 This leads to the idea that the human mind cannot therefore be a mere machine or a computer.
00:16:44.800 Because as a human, it seems to be inconceivable that one could be unable to judge as true
00:16:53.840 This is an often repeated mantra of some who write popular science.
00:16:57.840 At best, it is overreaching, while at worst it's disingenuous.
00:17:02.640 The fact is we can produce any number of statements that are true and cannot be judged
00:17:11.240 Barack Obama cannot consistently judge this statement to be true.
00:17:18.360 You can see the statement is true, but despite its truth.
00:17:22.400 Barack Obama cannot, however hard he tries, consistently judge it as true.
00:17:29.280 Girdle's incompleteness Theorems do not have any supernatural consequences.
00:17:33.400 They say something quite strange and surprising about mathematics, but this does not mean
00:17:38.080 that it then justifies everything strange and surprising about mathematics.
00:17:43.680 It is strange and surprising that sometimes very smart people draw conclusions from science
00:17:51.720 Science and mathematics are fascinating, and we need not draw any conclusions about some
00:17:56.200 of the amazing results we encounter in those fields unless we are compelled to by the evidence.
00:18:00.640 Be that scientific or in this case, mathematical.
00:18:04.120 So there we have it, a talkcast about the limitations of mathematics.
00:18:08.360 They will always be mathematical statements that even the techniques of mathematics won't
00:18:14.400 It's not all mechanistic and impersonal, it's fascinating and requires a high level of
00:18:18.440 creativity, and the only thing you should expect is to be continually surprised the
