Turings Universal Computer

this is a talk that i originally gave with greg michelson at the centenary of during's birth for the british mathematical colloquium it's off the topic of what i've been covering recently it's more hard science in particular it's the presentation of materialist mathematics a materialist interpretation of mathematics and the role that turing's idea of the universal computer played in the development of that so why universal well there were predecessors there are lots of computing machines before turing and these were put to practical use and why digital well there were digital machines before this is the anti-kera mechanism it was an ancient greek technology to predict lunar eclipses and lunar positions probably based on the work of apple polonius it could perform fixed ratio digital multiplications and additions using cogwheels fixed ratio and digital because the cog wheels have discrete positions set by their teeth and fixed ratios of numbers of teeth there is a model built by tanya van vark of how it probably worked we say it implemented apollonius's eccentric orbital model because the inner mechanism reveals this apollonius's model was a competitor to ptolemy's model or the the standard greek epicyclic model um the epicyclic model of the moon's orbit is that it this line is the earth there's the moon you go off to a center point which goes round and the epicycle goes around that center point apollonius's alternative model is to say that the moon rotates in a circle around point d and point d itself orbits around the earth we can compare that to kepler's idea kepler's idea is that the earth and moon are involve an elliptical orbit with the earth at one center of the or one node of the ellipse in apollonius's model the earth is here the the moon is rotating around this point which itself rotates around the earth and the actual observable difference in the moon's position between looking to here and seeing where the moon appears to be and kepler's one is quite small now we say the thing was built to implement this because it contains a mechanism like this there's a wheel b driven by a and as b goes round there's a slot in it and this slot goes round and the slot causes c to rotate at a different rate as it moves around so this goes round here and a pin d on disk c sits in this slot and therefore the orbital velocity of c varies according to the position of its rotational position of the wheel b now before turing no analog computers let's take an example 100 years ago navies had to solve real-time vector arithmetic problems and regression problems so way beyond what could be done with pencil and paper basically they were trying to work out fire control gunnery problems you've they had ships firing at other distant ships maybe 10 kilometers away and both ships were moving and they had to try and hit the moving ship ranges were 10 kilometers 10 miles relative velocity is about 60 miles per hour time of flight of the order of 20 seconds positions were found using bar and stroud rangefinders um baron stroud had the original works in ashton lane next to glasgow university um it's a matter of some interest to me since a lot of my work at glasgow university was involved in stereo photogrammetry and 3d tv studios which used basically the same principle as bar and stride we're using of stereoscopic estimation of range okay here we've got two ships there's a bearing of b relative to ship a b has a velocity um which we say in vector format 15 minus three a has a velocity 0-14 i is moving 14 knots straight down this one is moving 15 knots west and 3 and 3 knots uh south and you need to form a vector sum of these two quantities to get the velocity of b relative to a which is 15 11. so you have to perform vector arithmetic to get the relative positions and the relative velocities and you've got a bearing you have to predict where that ship will be in its relative bearing 20 seconds later or however many seconds later the actual fall of shot is going to take now this is a picture of um commander john dumaresk of the royal australian um navy who's considered by his college to be a mathematical genius and is an unsung pioneer of computing he doesn't often get mentioned or a pioneer of mechanical computing this is the device he invented now it doesn't look like a computer by the sense that we now think of computers it's really weird device this line here is aligned with the bow of the ship the arrow here points to the bar of the ship so this device is fixed on the ship that points to the bow you rotate this disc here so that you have a sight bearing on the enemy ship you adjust this moving mechanism here forward until it points this pointer here points to the number of knots that you are are making how many knots forward are you making you rotate this dial to your s this pointer to your estimate of the course that the enemy is sailing and then you read off on this dial the distance away that the enemy sorry not the distance away the speed the enemy is going and where you get a a point here you can read off on this scale the rate at which the enemy's range is closing or widening this is a closing range and on the other scale you can read off the deflection um forward that the enemy's ship has so this mechanical device carries out that piece of vector arithmetic and it does that piece of vector arithmetic by mechanically implementing in an analog fashion the vector operations now that's not the kind of thing that turing wanted turing was trying to deal with the decision problem which is the problem of whether there is a mechanical procedure for the decision as to whether a mathematical theorem is deducible from its axioms now one of the factors that turing needed in order to do this was an ability to have self-reference now at an earlier stage godel had handled a similar problem and had to devise an encoding of of formula and theorems which allowed self-reference in terms of arithmetic of these problems turing's genius was to incorporate this self-reference in another way but this property of self-reference which was incorporated for reasons of mathematical proof was the key to the economical adoption of computer technology gives universality what turing said this special property of digital computers that they can mimic any discrete state machine is described by saying they are universal machines now this concept of universal machines is very important actually i think provides a key to understanding the concept of abstract labor in another context now universality brings cheapness the universal machine has the advantage that one design can be applied to any problem this brings a huge economy of scale to manufacture millions and millions of intel and arm chips can be turned out all identical but all being applied to different problems so let's look at the digital aspect i'm quoting turing now the machine is digital how that the machine is digital has a more subtle significance it means firstly that numbers can be represented by strings of digits that can be as long as one wishes one can therefore work to any desired degree of accuracy this accuracy is not obtained by more careful machining of parts control of temperature variations and such means but by a slight increase in the amount of equipment in the machine to double the number of significant figures would involve increasing the amount of equipment by a factor definitely less than two and would also have the same would have some effect increase in increasing the time to take each job this is in sharp contrast with analog machines and continuous variable machines such as the differential analyzer where each additional decimal digit required necessitates a complete redesign of the machine and an increase of the cost by as much as a factor of 10. the differential analyzer was actually an outgrowth of various aspects of that have been previously used in naval fire control computing turing's argument here is very pragmatic and there's been a recent temptation to think that we can outperform digital computation by reverting to analog computation some of this is referred to as hyper computation in the literature and it has actually been utilized by economists of the austrian school to say that problems of economic planning are so complex that they would require what's called hyper computation we're very dubious of the idea of hyper computation but the the very idea of using analog computing to overcome what can be done with digital computing is based on the philosophical misconception the misconception is that in reality everything is continuous but we know that this is in fact false everything is digital or quantized the notion of the continuum arose in classical greek geometry from the proof of the irrationality of the length of the hypothesis hypotenuse of a right triangle with unit sides if one assumes as the greeks did that classical geometry was a true theory of the real world this implied that space itself must be continuously subdivisible but how do we know that pythagoras theorem actually works in the real world could we even experimentally test it and this is a hypothetical set up with half silver mirrors to verify pythagoras theorem suppose you knew that ac and a b had been previously measured to be an integral number of wavelengths of light now if one was able to show that the complete path a c b a again was also an integral number of wavelengths of light one would have therefore disproven pythagoras theorem but there's all sorts of practical problems to this you you you can't simultaneously measure bc and ac because of mirror orientation and you can't know whether you've actually set up a right angle without pre-assuming pythagoras theorem because the practical techniques you use to set up right angles assume it more generally though the fundamental limit to spatial accuracy of any computing device is provided by the plank length which is 10 to the minus 15 meters which limits the fundamental accuracy of any even hypothetical analog computer and most proposals for trans during computing are based on the illusion that real numbers are real in the ontological sense they're based on continuum models of the world like maxwell's equations or newtonian mechanics in the post during era we have to see theories like maxwell's equations or newtonian mechanics as essentially software packages for making predictions about reality when we combine this with the computer do the mass they allow us to build models that mimic some part of reality but just like the anti-thera mechanism there are limits to the accuracy of our software packages and our software packages are not reality itself suppose using some continuum model of mechanics we showed the system with computable boundary conditions has some points where the parameters are uncomputable does this tell us that the real world can do things which are uncomputable no it doesn't i give that as an example because examples like that have been cited as proof that computation must exist no it doesn't it tells you the software package or physical theory we're using as a bug in it and it's just such a bug in maxwell's equations called the ultraviolet catastrophe that let einstein invent the quantum theory the point is that reality is digital and during computability rules now why a computer this is the console of the actual computer that turing built the pilot ace he wasn't just a theoretical mathematician he actually built computers now it's often said that the turing machine is equivalent to the lambda calculus of alonzo church now is the universal computer equivalent to the lambda calculus no hold on as a controlled experiment we took a simple lantern expression here and left it overnight on top of a book with the relevant theory but it failed to evaluate but we put it into something called the lambda can it's a little box here's the formula same formula lambda x x plus one two put it into the lambda can and we get back the answer three which is what the lambda calculus says you should get back but there's a missing secret ingredient in the lambda cam a universal turing machine or at least a universal computer in the turing sense the lambda calculus is only equivalent to the universal computer if by the lambda calculus will mean either a lambda interpreter on a universal computer or a mathematician a blackboard and a definition of the calculus that the mathematician understands both of those are material configurations turing brings out the importance of physical embodiment of calculation by introducing a machine he introduces mechanics and indirectly physics as a support for mathematics talk was from me and from greg michelson you

Channel: Paul Cockshott

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Length: 20min 21sec (1221 seconds)

Published: Fri Aug 07 2020