 this will be scary or sad. hallway. And I'm going to be sad. Yn ystafell, Tom Hall. Mae'r Tom Hall. Mae'r hwn yn y internetiaid. Mae'r Tarsea. Mae'r tawch yn y tawch yn y tawch, mae'r tawch yn fawr, mae'n ddweud o'r tawch. Mae'n ddweud o'r tawch yn y tawch ymlaen, mae'n ddweud o'r tawch. Mae'n ddweud o'r tawch, mae'n ddweud o'r eich hwn, o'r eich llifth o'r tawch, o'r... не sy'n golygu a dylai'r bydd y bydd. Mae'n ddweud o'r tawch o'r... ..o'r cymran, mae'n ddweud o'r ffurth o'r bydd. E'na grai, mae'n ddweud o'r ffurth o'r busund. Mae'n ddweud o'r busund, mae'r gweithio'r feddwl, ein ddweud o'r 15,000 erbyn. A mi ddweud. Ymlaen yw hyffurdai o ffyrdd ymlaen i ddenedigol, mae'n ddweud o'r sgfeddus Dydydd ni ddaeth yn wneud uwch fel y gallwn. Mae'n bod yn lineil yn fawr y Cysylltiadol, a fyddai'n ffordd arall y cyfrifigol. Fel dwi'n deall, mae'n byw'r cyfrifigol, a fyddai'n wybod yn gwybod. Felly mae'n teimlu i gyflod i gyd yn y maen nhw sy'n gyd byw'r gweld yn gweld i leifiau drwy'r bwyddyn, a mae'n ddysgu amser osu hynny, rydyn ni'n gweithio fydigol i'r ffordd. A oes, dwi'n ddim yn gweithio'r cyrfi, that's not previously possible so I want to just talk about a few creations, discoveries and then how they're used and how that might be relevant to software development or programming or computation, so if you'll bear with me. Ond yw'n cefnod, y pethau'r idea'r greu. Ac mae wedi cael ei ffordd o'r rhaid i'r gwleidio agri-cylwyr am y cyfeirio. Mae'r rhaid i'n cefnod'r gweithio'r gweithio. A'r gweithio gweithio'r gweithio yn y ffostidliadau i'r gweithio'r gweithio, ond yn dweud hynny'n gweithio'r gweithio'r gweithio'r gweithio'r gweithio'r gweithio'r gweithio'r gweithio. Felly, ysgrifau, Or, I don't know if they would be called scribes in Mesopotamia, actually, but this is Cunea form, so what's considered to be the first script, it was used using wedge-shaped markers into clay tablets, which were then baked, so it may just be that they were the most durable form, so we have to be a bit wary of thinking people were obsessed with burrian people and obsessed with stones when they may actually have had quite a rich material culture that just doesn't survive. As always of archaeology, you have to be wary of the survival bias of the sort of durable objects. Writing was a big idea in, I think, the Mayan civilization, it was actually not, so there were time nots and some people running through the hills between the towns to try and send messages and stuff, but again, not everybody was doing this, just the rulers are using it. But, and I think relevant to lots of us, the people who were in that position, who had the special skills, firstly, it's quite often people who are already rich, you can afford to train their children in the thing that is important to pharaoh or whatever. And, as I say, it becomes an instrument of power, it's how pharaoh measures his taxes, it's how he declares war in large part, it's how he celebrates his victories, or if you've studied the ancient Egypt, they always celebrate their victories, and you have to, you can detect their defeats by the victories, become closer to home again as they get sort of chased out of other areas, which is quite nice. I don't know why I remember that, that's not relevant to my argument. So, yeah, writing was the great idea, at first it was an elite feud that did it, they were special and they were treated, and it was an instrument of power. This is a Lindisfarne Gospel, you can see them in Lindisfarne or the British Library have a few good examples. So, these are sort of many many man hours per page, and it was man hours then, I'm not being sexist, they were. So, many many man hours into them, again rich people can afford them, a special priestly class deals with the creation of these artefacts, and again, I don't want to say instruments of power because I might upset the religious among you, again though I would say access to them was limited. Not necessarily the case of some of these, some of these would have been devotional objects shared in churches and stuff, but certainly the spare capacity to afford their creation was only for the wealthy. Oh, I've got a note to myself because I can't necessarily guarantee that the pictures will trigger everything. So, I found out quite recently, just in terms of, we talk about privilege and stuff like that, at one point if you were, if you could recite Psalm 51 in Latin, you were tried by an ecclesiastical court, not a civil court, and they were much more lenient. So, when you look now at the differences of crack cocaine sentencing versus powdered cocaine sentencing and the societal connotations there and the colour of your skin correlating with how severe your sentences are, how likely you are to be stopped and searched, et cetera, actually being able to, well they said read, but really if you were smart enough to remember the sounds, you could do it. And I always read a chap called Gordon Child on this stuff, so sort of early civilisation from a sort of moderately left wing perspective. He's been called a Marxist archaeologist, but I don't think that's the case. But I discovered during the more recent last few days' worth of research a chap called Harvey J. Graff. Now, that's just more a note to me and to yourselves that it might be worth taking a peek at him, more contemporary historian of sort of, I guess, cultural historian. I am not a religious scholar. Oh, by the way, sorry, I added this claimer that I nearly added. I think to sort of paraphrase Brendan Eich on JavaScript, anything that's good in this talk is probably not original. And anything that's original is probably not good. So full disclaimer, yeah. And also, yeah, I'm not a religious studies expert. And then the printing press came along, democratised things. But as you probably know, there was a battle around that as well, who gets to control them. Is it illegal? And there are also large, expensive devices. So then it's very likely that you are a wealthy to-do person before you can get one. So people club together, they start using them, then they're made illegal, et cetera. So again, there's always battles around technology. So there's people here who are thinking to talk better about that. Coincidentally, while I was preparing this talk, I started reading a book called The Intellectuals and the Masses. So this is a chap called John Kerry, was invited to give a very prestigious letter series on, I think a single letter on modernism, and basically slaughtered the whole endeavour. I'm not saying I agree entirely with his premise, which seems to be that as mass literature arrives, so from, say, 1850 to 1900, you start to get mass literature. By 1920, most people in Western Europe can read. And the intellectuals say, well, what makes us special now? Why? If everybody can do this, we can't allude to books we've read and we can't be the ones that have the secret knowledge. So his argument is that essentially literary modernism, this sort of what seems to me impenetrable work, which I'm conflicted because I have good friends that are really, really into it and really value it. So I don't want to say I'm sort of all in with Kerry here, but I think the book does paint the actual, and I know it's a bit ad homonym, but the actual characters doing this work are pro-fascist, pro-Euthanasia, or what's worse than Euthanasia? My mind's gone a blank. The one way you just want to kill lots of people, not necessarily people that want it. No, I'm thinking eugenics, they call it, didn't they? Good genics, yeah, sorry. Excuse me, my memory's a little off. I'm a little bit unwell. So, yeah, his argument is that that is the point of it. The point of it is to exclude people. So there's another book by a chap called Hoggart called... Shouldn't try and do things off the cuff. Uses of Literacy it is, and he takes a different view to this where he thinks that the masses are kind of being dumbed down, whereas this is, I guess, maybe an opposing or opposite view that the intellectuals are sort of retreating somewhere else, which I thought was quite interesting. So, and yeah, and it was actually, D.H. Lawrence explicitly says that they're trying to create a body of esoteric doctrine defended from the herd. So, the point of it is almost to exclude people. And I worry when I look at the discourse in communities I'm part of and let's say someone discovers Haskell, they're really, really happy that they found this new purely functional thing. And then you talk to them about somebody who does PHP or whatever and they're like, oh, no, that's disgusting. And I don't know if it's... Are we trying to bring everyone along to the nice place where we are? Are we trying to say, I've found something great, guys, come on, pile in, let's all go and have a look at this? Or are we saying, I kind of like that I know what a homomorphism is and are we saying that, you know, oh, I like the reserved words and I'm not arguing that people are. Well, I'm not arguing that all people are necessarily, but I think there's a danger of it. There's a danger of finding something new and sneering at the masses that are left over. Oh, still doing Java. Oh, ooh, et cetera. So I just wanted us to be aware of that. One thing... Sorry, this is a bit ramboli. So excuse me. I'm not going to change it. I'm just going to apologise in the middle. What do you say? Hat, was that? Oh, sorry, I thought someone said... Excuse me. I thought someone said hat. And I was trying to figure out why that looked like a hat. So, yeah, so mathematics is another area of interest of mine, my degree was in mathematics. So this is a Babylonian clay tablet again. Again, using the same wedge shaped things to make the indentations. It was much easier to be a polymath back then. So the people that were the writers were also the mathematicians, were also in large part the philosophers. That becomes less true over time as we kind of branch out and find our little niche in order to take knowledge forward and get our PhDs, et cetera. So this is actually, I think, I stole it from Wikipedia and it's somehow depicting the square root of two, which they did know about, which is quite interesting. So now I'm going to talk briefly about mathematics. Mathematics has a similar kind of path where it starts off as a thing just for the elite. Then it's an instrument of power again, like how much taxes do you go? How much interest do I apply to that loan? You know, the wongas of the sort of bronze age or whatever. And then now everybody's sort of kind of does it. Now it isn't as true as literacy that everybody really does mathematics with the fluency with which they do read and write. Now I was a maths teacher for a while and when I say that it's the fault of education, I mean it's my fault as well. But some things are, so sometimes things enter into the vernacular and you don't really know they have. So when I was a kid, I used to play top trumps and one of the numbers that you could, everyone knows how top trumps work, right? You choose a category or a trait or something and you compete on the numbers high and low, etc. One of the top trumps stats was 0 to 60 and how many seconds it took. Now that idea is a compound rate of change, isn't it? It's a second order differential. That is actually unthinkable, sort of pre-calculus. Acceleration is actually in itself quite a complicated topic. But people now that drive cars do understand it. So even though they don't think they're doing mathematics, there is a conversational fluency with what is quite a complicated idea, like a second order differential. I mean, people thought for a long time that heavier things fell faster and things like this and gravitation. The idea is to take a while to arrive. Actually, I noticed when I searched for this image and I don't know if this is the dumbing down of education but that is no longer on the car image I could find. So as I was preparing this talk, I think I've got via osmosis something from this chap at IBM Watson, the research lab, I think outside New York. The article, it's more of a book in fact, is about notation as a tool of thought. So when I first studied calculus, I thought there was a certain things that are easy because of the way the notation is chosen. Apologies for the iffy image. You know if you say dy by dx, that's the rate of change in y with respect to x. If you actually have it in terms of another variable, this equation is true. People think it's obvious because all the dts just cancel like a fraction. Actually, when Newton, you might know Newton and Leibniz, not co-discovered as in working together because I think Newton was a bit of a... Well, he wasn't a very nice person from what I've been telling. So, yeah, it looks obvious that dts cancel like you would a fraction. In reality, Leibniz, absolute genius, chose this notation because these rules work. One over dy by dx is dx over dy, if you remember fractions. It's exactly the same thing. Dy by... This is the right way round off the cuff. Dy by dt divided by dx by dt would be dy by dx. Again, you divide by flipping and then multiplying. The vision to see that this thing had the same properties that people already know from fraction and choosing a good notation and letting the notation do some of the work was, I think, a stunning bit of insight by Leibniz. More recently, I've been studying quantum mechanics. I've never actually did it at university. I thought another similar, very nice example was we've long had this notation on the left for an inner product. So, a and b are vectors. The inner product, you might see sometimes you write a dot b and that's basically a scalar quantity related to the two vectors. Dirac had an insight with this operator of separating it out into a bra and a ket and using the properties of those two operators, like how they distribute, et cetera, chose a great notation, so the notation actually helps. There may be better quantum physicists than I in the audience. They're perhaps one of a drug people at the end and go through exactly why that's awesome. So there I kind of liked it again. Choosing a notation, you put those two things next to each other. You apply that operator on the left to the operator on the right. You get the inner product, which is just a scalar. But then you have also, as I say, distributives of laws and things like that within them. So, if you have a bunch of these things, you can group them together because they are rather associative. You can actually take them in pairs and cancel them all down. Again, so the notation is kind of working for you. So, yeah, the quotes I wanted, and I saw it in SICP, so the Jerry Sussman, I think it might be a jagstra quote, where he says, we're so early in the age of computer science that we confuse the tool with the topic, so we think it's about computers, and it's like calling biology, microscope science. So there's a microscope scientist doing some work. I think animal welfare charity. So, yeah, this idea that we've confused the tool with the topic. So, how did the tool, well, how did the topic come around? So, at the turn of the last century, so like 1900 or so, there was a programme to formalise all of mathematics. Now, there's very good reasons why we should be happy that people didn't succeed, because, well, without girls' theorem, mathematics would by now have been very boring, so nobody was thinking about computers as we have them. They were just thinking about ways of guaranteeing things were true by starting with axioms and then using only symbol manipulation, using only explicit rules that are also, I was going to say machine checkable, but that wasn't quite what they said, that were checkable by some procedure. So this is from a textbook I used, and the idea is your rules you're using are on the right, and then you're in the world of only manipulating symbols. Now, the point of these formal systems is anything you show in the formal system, anything you derive, is actually true in the real world. So a proof, which is just something you find in the formal system, corresponds to a truth, and the hope was that we could get to all mathematical truths by choosing enough assumptions and the correct set of rules. Now, no set of rules is adequate, which is great, because it means mathematicians will be in a job for a long time. But the problem was they're talking about, so every step here, you've got multiple choices, there's loads of different rules you might use, some goal, it's directed by a human at this point, so at this point you make assumptions, but you've got a goal in mind, you use the rules, but you've still got an end in mind, and at some point you say, well, I'm done now, and then the property of the formal system is every line of this proof can be checked by a dumb procedure, but this is still sort of requiring insights. So the next step is trying to express procedures unambiguously, so the two most common ones are the Turing machine and the Lambda calculus. Hopefully if this refreshes, I had some trouble uploading my... I've spoiled a joke now. And I've lost my picture of the Lambda calculus. So the Lambda calculus, from an initial starting point, it evolves deterministically. At every stage there's only one thing you can do, there's only one thing you can do, there's only one thing you can do, and then it ends, so it's much more like a computer program. So we go from, and that's to be very hand-waved, on the Lambda calculus, but I'm afraid. So here we say human sets up some assumptions, human uses the rules he knows about to get the result he wants, and then says I'm done. The Lambda calculus is, here is some expression that satisfies the rules. Every step we say, apply the rule, apply the rule, apply the rule, apply the rule, I can't do anything anymore, that is now terminated, we're now done with that, we have the result of doing that computation. Turing machine is very similar, move up and down the tape, got some rules, read and write from the cells, move up and down, and at some point you know that you're done. So what time are we on? Sorry I can't see o'clock. 24, okay. So, yeah, I was going to say, we've got more computing power in our pockets than anybody realised, and I think everybody thinks, yeah, my iPhone is really powerful, but in reality almost anything would be more powerful than they realised. Doing these things by hand is almost impossible. So, I wanted to talk now, so that's kind of how computers arose. They arose out of mathematical logic, then suddenly we put them in a dumb device that does loads of things really quickly, and then suddenly we arrive at like, the computer revolution, if it can be called that. So, but now we can use them to do other stuff, so given that they exist, and again, I meant to say before, we now, we always talk about programming languages and stuff, so we always say, we have like, you know, the Haskell meetup or the whatever language you like to meet up, but then, I don't know if in ancient Egypt they had like the particular pen meetup, the kind of papyrus meetup, and whether they were so invested, like who has the best papyrus, oh, you use the slightly yellow one, I use the more browny red one, you know. So, I don't know whether that's the case, but given that these things exist, how can we use them, and is it done to use them, or is it smart, or is it always with older mathematics teachers, or that's less common in fact, being unfair to them, with older people who have studied mathematics in the past, and who are, are mathematically much more efficient than I am, I think using a tool to sort of save you the effort is really good, and certain other things become possible previously, so we can say to a kid now, for instance, what happens if you take a number and keep taking the square root, where does it go? So you might say they choose 9, keep pressing the square root, where does it go, and the answer is, and this calculator does behave like the ones I had. You basically overflow, and you can't tell a difference anymore, and at some point it's just 1. So there's some accuracy that it doesn't display, and it's just 1. That's a really interesting discussion, so what do you say to the kid to start with a really big number, and you keep pressing the square root what happens? It's good for your intuition. I'm not doing any calculation, but I've got a great intuition that for one thing, taking the square root makes the number small really quickly. For another thing, this iterative process, after however many iterations converges on one really quickly, for somebody before it to really get, and there's a great question now. I've just gone through, obviously not exhaustively, because the real numbers are quite a lot of them, but I've gone through quite a few examples now, and I'm developing an intuition, so a good question might be, well, what would be a good number to try? You know, a kid's got some good intuition, maybe he will say, well, what about a number less than 1? What happens there? Yeah, cool. So what about a number less than 1, so 0.002? Does this have the same behaviour? What does it mean? It's getting bigger, and then again, it converges on one from below. So you've got basically a dynamic system. You start up here, it heads towards one, you start down there, it heads towards one. A great next question, or maybe they said what happens with one, and he's like, great, and then if a kid's very smart, they might say, well, what about minus numbers? And then you're straight away into a very interesting discussion, aren't you? So you're straight away into a very interesting discussion, but you're not doing any calculation because the dumb machine's done it for you really quickly. So that's kind of what I really enjoyed about the fact that you can use computers not for, or not talk about computers, I'm not talking about the JavaScript that implements this. It's a machine that I'm using to illustrate some other points. So I thought that was rather nice. So let me do that for you. All of A-level mathematics, certainly, and a good slice of degree-level mathematics, in fact, can be solved by a mathematician now. So if you really do want to solve problems, use the tool, get those really dumb machine churn and really fast to do it for you. Oh sorry, I've gone off the full screen. So leverage them wherever you can, leave the insight that's necessary for you. So one final demo was, again, developing an intuition in a way that is completely impossible, impossible, impossible before computers. So if you ever did any pencil and compass constructions, they're actually really hard, and especially if you're practic or whatever, you have like attention problems because it actually takes ages. It's actually fairly boring in the process. The revelation at the end isn't actually that good. And you basically you basically take, I don't know if you've done it, if you might remember, you take a shape a 2D shape and you can enlarge it, transform it, rotate it, etc. So one thing I absolutely adored was some software called Geo, well I had a non-free version. There's a free version called GeoGibra, which is great. What's let you set up, can you see the origin here, the little green dot? And then the F is the thing I'm going to rotate. So if I fix the origin there and then rotate through some angle, what does it look like? So here I've got a sort of dynamic view of what it's like, so at zero if I rotate none and then I start going through it, that the extra intuition you get from knowing what roughly where's it going to be, etc. You can't really get with a compass construction or a ruler construction because it just takes ages. Similar for reflections. Actually getting a feel for what reflection is as an operation is actually quite hard and you can kind of mechanically follow the steps, you know, go to the mirror, go out at the same angle, etc. But to get an idea intuitively of what it means, if the mirror becomes very close to the object or I think I can move the object as well, I can't in this one, or the object becomes very close to the mirror, what happens there, does that really make sense and like they can do things that perhaps you wouldn't do otherwise. Yeah, two more of these because I like them. I don't know, it's not working. Oh yeah, cool. So this one is brilliant, so enlargement via a scale factor. You might remember in the bottom left corner is the sort of centre of enlargement. Here's my shape. So if I change the shape, the enlarge shape changes. That's really good. This would be a ruler line in the normal case or the pre-computer case but here I'm just moving it around to build up my intuition. So if I change that one, that one changes. If I change this one, obviously this one changes. And that's really good for developing an intuition and what does the scale factor do, so you're always, if I was to construct this with a ruler, I would basically take that point, draw lines through the points I'm interested in and I would measure the distance to the first one and I would say, oh well, if I want to scale factor three, I'll go three times as far and then the image that's drawn is three times as big. So sorry if you perhaps don't remember the school mathematics but now what if we change the enlargement factor, so really really nice is scale factor two, it's twice as big, scale factor three, it's three times as big but further away you see how we're going along those rays. And really nice is, well you can ask kids, well what do you think scale factor one will be? And it's actually a really easy question asked like this, you say, well what's scale factor one going to be? And they go, oh it'll just be the same. That's a really hard insight to arrive at actually. And you go, well what about a fractional one? And it's an almost impossible question until you've seen this but if I say now, what's one going to be, what's fractional going to be? Actually it's much easier to get, a fraction of a half if being closer to the centre of enlargement and half the size is actually really good intuition. And one more, which is the most difficult one is bisecton align, so this was known by Euclid. You've got two points here, A, B. I've drawn a circle more than half of the distance between them, the same size from each. Where those circles intersect, I've drawn another line. That line is a perpendicular bisector of the line A, B. So I'm sorry if you can't see that very well. But the great thing again for developing that intuition is well what if A and B were further apart? Oh well it still works doesn't it? What if C was shorter, so my circumference of the circles was shorter and again straight away you visualise that it only works if I go at least half of the distance. But as long as I go the same distance on each side the bisector works. So that's really nice. Rather than say to the kids you've got a rule you have to go at least half way you draw them and then it bisects the line, they can see why then. They know if it doesn't reach quite half way that the circles don't intersect at all and then you can I think that's obviously move the circle in, make the line shorter. I think that's really nice. So that's basically using the output of programming or coding or whatever to get insight into some funnels. One other thing I've been interested in recently is using the actual method itself so using programming using the fact that we can unambiguously express procedural knowledge to learn other things. So Jerry Sussman and Jack Wisdom I've got a whole introduction to classical mechanics using the programming language scheme rather than normal mathematics which I thought was an interesting idea. There's introductions to Bayesian statistics and statistics using Python which I kind of like, it's like given that you know Python let's learn statistics rather than perhaps the other way around so I thought again that was interesting using the fact that you know programming to learn something else. So that was a little bit meandering so thank you for your indulgence. So my only conclusion is I guess are we being esoteric or are we inviting people to these things that we've decided objectively as beings of pure rationality that are better than the other things. Are we inviting people in or are we kind of glad that we're in an elite group of people and what happens when everybody does it? I think it's an interesting problem. I think it's an interesting idea an interesting premise and if it gets to the point where it's like writing where everybody just does it as a matter of course oh yeah just program the car do whatever I think that would be awesome and the idea we can use the output of this great idea, this computation to support other stuff but we can also directly use the idea given that people already know a programming language can we use it as a medium of explaining it the same way I would use the fact that I can read to learn about churches in Gothic France or something. So yeah thanks very much.