On Monday February 4th, I gave the first of a series of three free public lectures on computng and mathematics at Gresham College, Barnards Inn Hall, Holborn, Central London. Any reader of this blog is welcome to attend these lectures.

The first lecture was entitled "Arithmetic by Human and Arithmetic by Computer". Video and audio recordings are available at the Gresham College website. This lecture featured some cool tricks for doing arithmetic, a performing monkey and brilliant Scottish inventions.

The second lecture was "How Computers get it Wrong: 2+2=5". Video and audio recordings are now available at the Gresham College website. This lecture discussed different kinds of computer error.

The third lecture will be given at Gresham College (Barnards Inn Hall, Holborn, central London) on Monday 15th April.

Please post any comments or questions about the lectures as comments on this blog post.

Dear Prof Mann,

ReplyDeleteThank you for the Mon 15/04 lecture which I attended and enjoyed.

Your lecture brought to my mind several topics over which I thought in years gone by.

The topic I would like to discuss here is the following:

Like many other mathematicians I heard before, you mentioned that mathematicians like to keep mathematics "pure" - they would not use a result before it was rigorously proved to be true.

There is a counter example which is hardly ever mentioned in this context:

When Leibniz and Newton introduced the calculus in the second half of the seventeenth century, they introducecd a "phenomenon" I will refer to as an infinitessimal (d). This d has two seemingly contradictory properties:

a) It is not zero and, hence, it is permissible to divide by it and

b) When added to a real number, it is deemed to be negligible (a zero).

Some mathematicians and the philosopher Berkeley strongly objected to this new phenomenon fearing that it may incorporate a "time bomb".

Other eminent mathematicians and physicists took advantage of the calculus to produce some wonderful results knowingly oblivious to the warnings sounded by Berkeley et al.

It took some two hundred years before the mathematical community accepted that it was safe to state that the derivative of x^2 is, indeed, 2x. This was done by jettisoning d and introducing sequences and limits.

Interestingly, the infinitessimal reappeared in the 1960s.

The purpose of the above is not to "show off" but to demonstrate that mathematicians are not always "pure".

Best regards,

Victor Nicola (victor.nicola@talk21.com)

Victor,

ReplyDeleteMany thanks. You're absolutely right in your account and what I should have said is that *some* mathematicians are reluctant to use a result before it is proved. Many users of mathematics are happy to use any method that works.

Raymond Flood gave an excellent lecture at Gresham College last term on the history of the calculus - worth watching the video if you haven't already seen it!

Many thanks for your interest, Tony

Having only just read the content of your lecture 'Proof by Computer & Proof by Human' which I found very interesting, at least those parts of it that I could assimilate, I would like to make the following observations.

ReplyDelete[1] My mind fails to see any gap in Euclid's proof of his Proposition 1 since the construction of two circles of equal radius, the second of which is centred on the circumference of the first logically results in their intersection at 2 places so that any additional embellishments to prove they intersect are in my opinion and I would suspect that of most other persons, simply superfluous.

[2] I totally fail to understand all the hype surrounding 'Kepler's Sphere Packing Problem'. In an infinite array of spheres in the close packed hexagonal configuration any 1 sphere is in contact with 6 spheres in a hexagonal array around it's equator and 3 contacting spheres around its north pole and a further three around the south pole, a total assembly of 13 contacting spheres. Now all the centres of the contacting spheres can be joined by a straight line passing through the point of contact equal to the diameter of the spheres. To increase the density of packing a 14th sphere would have to be introduced so pray tell me where? To me it beggars belief that after 400 years it takes a computer using 5000 configurations to be just 99% sure that the closed packed structures are indeed the best possible assemblage. Where on earth did he get 5000 different configurations from when there are at most 7? basic arrangements for the atoms in any crystal found in nature and atoms are virtually empty space and the close packed hexagonal /face centred cubic are indeed the densest assemblage!

[3] That Lois de Branges claim to have proved 'The Riemann Hypothesis' made it into 'The Guardian' newspaper where a celebrity Oxford professor was sceptical that it was indeed the proof. He considered that the proof of 'TRH' could eventually result in a 'prime spectrometer' and bring e-commerce to its knees since security base on RSA would be compromised. As security has not yet been compromised do we take it that Lois de Branges has not proved 'T.R.H.' and is this then 'proof/ disproof by consequent outcomes' or do we need to wait a bit longer before drawing a conclusion?

[4] Finally what should one do if they find a simple proof that everyone says does not and cannot exist yet every fibre of their being tells them it is correct? Seek psychiatric help, commit suicide or seek the opinion of a mathematician?

For example if we take the following algebraic identity z^n = y^n + x^n we can ask the question what values if any of n result in an all integer solution.

[Step 1] Multiply the equation by z^n to give (z^n)^2 = (z.y)^n + (z.x)^n

[Step 2] Multiply the equation by (z^n)^2 to give (z^(2n))^2 = (z^3.y)^n + (z^3.x)^n

[Step 3] Multiply the equation by (z^n)^2 to give (z^(3n))^2 = (z^5.y)^n + (z^5.x)^n

[Step 4] Multiply the equation by (z^n)^2 to give (z^(4n))^2 = (z^7.y)^n + (z^7.x)^n and so AD INFINITUM.

Now every (integer)^2 can be split into the 'difference of two squares' and re-arranged to produce a 'Pythagorean Triple' which for the infinity of odd integers are all primitive (Pythagorean series) and for the even integers are primitive for all (integer)^2 MOD 8 = 0 (Platonic series).

I can therefore only draw the conclusion that the equation z^n = y^n + x^n has a one to one correspondence with the 'Pythagorean Triples' so the only conceivable value for n is 2 and so any higher value for n would be an absurdity . QED.

I have loaded the bullet into the chamber.

I have spun the chamber.

There is a bullet in the firing position if the proof is FALSE.

There is no bullet in the firing position if the Proof is TRUE.

CLICK!! AM I DEAD OR ALIVE.

Alastair,

ReplyDeleteThank you for your observations.

1) I agree that the existence of the intersections is obvious, but it doesn't follow from Euclid's five axioms. But I don;t think anybody seriously worried about this for a couple of thousand years!

2) It's a long time since I looked at the Kepler conjecture so I can't really comment.

3) We need to wait longer: de Branges' proof is sketchy and as far as I know has not so far been disproved.

4) Again not my area, I'm afraid.

Thank you for your reply Tony. Much appreciated.

ReplyDeleteBrief Bibliographical Request

ReplyDeleteDear Professor Mann:

My name is David (from the States) and I have taught English for several years. Thank you for your Gresham lectures. I listen to each one again and again and have shared them with my students. 'Calendar Curiosities', 'Heading for Trouble', and 'Arithmetic by Computer and by Human' fascinate me perhaps the most.

Could you share the source of one of your audio clips? Beginning around 23 minutes and 56 seconds into the audio file hosted at , you play an extract from someone reading Mark Twain's _Life on the Mississippi_ to illustrate the perils of extrapolation with a non-representative mathematical model. I'd love to buy the audiobook, if that's the format, or otherwise locate and share that particular reading with friends.

Regards,

David

Thanks for your kind comments David. The extract was recorded for the lecture by Noel-Ann Bradshaw.

DeleteAha! As soon as I realized those readings were meant for the lectures, I recalled having heard the same voice as Newton and a housekeeper in 'Arithmetic by Computer and by Human'.

DeleteThat lecture led me, by some intermediate steps, to discover _Dead Reckoning_ by Ron Doerfler, _The Great Mental Calculators_ by Steven B. Smith, and _The Reckoning of Time_ by the Venerable Bede.