Posted by: Alexandre Borovik | December 1, 2011

Expressive power of natural languages

I will follow discussion of that question on the Foundation of Mathematics mail list with great interest:

Alex Nowak:

I was wondering if there is any (at least semi-)conclusive view about the expressive power of a natural language like english resulting in a statement like “whatever it is, it is a language of at least 2nd order”. Of course, I know of Tarski’s comment suspecting natural languages to be somehow (semantically) universal. But what I’m interested in is a hint pointing me in a direction what to look for, i.e. is the fact that one quantifies over classes in a natural language enough to label it higher order? Can there be anything wrong to take it to be at least a many-sorted first-order language?

Read More…

Posted by: Alexandre Borovik | December 1, 2011

Can you crack it?

Can you crack it?

Can you crack it?

Posted by: Alexandre Borovik | November 30, 2011

Axiom of Choice

Posted by: Alexandre Borovik | November 29, 2011

Quantum Magazine on Google Books

Quantum Magazine gave Google permission to let the public view all the issues of Quantum they have indexed in Google Books. A few years are missing, but there’s a lot there. See direct links:

Ian Livingstone, life president of the British video-game publisher Eidos, published a article in The Independent. A quote:

The National Curriculum requires schools to teach not computer science but ICT – a strange hybrid of desktop-publishing lessons and Microsoft tutorials. While Word and Excel are useful vocational skills, they are never going to equip anybody for a career in video games or visual effects. Computer science is different. It is a vital, analytical discipline, and a system of logical thinking that is as relevant to the modern world as physics, chemistry or biology. Computer science is to ICT what writing is to reading. It is the difference between making an application and using one. It is the combination of computer programming skills and creativity by which world-changing companies such as Google, Facebook, Twitter and Zynga are built. Indeed, in a world where computers define so much of how society works, I would argue that computer science is “essential knowledge” for the 21st century.

Posted by: Alexandre Borovik | November 23, 2011

The shortest ever abstract of a research paper?

URL: http://arxiv.org/ftp/arxiv/papers/1110/1110.2832.pdf

Title: Can apparent superluminal neutrino speeds be explained as a quantum weak measurement?

Authors: M V Berry, N Brunner, S Popescu & P Shukla

Published: November 11 2011, J.Phys.A 44 492001

Abstract: Probably not.

Posted by: Alexandre Borovik | November 23, 2011

Open book examinations in the Internet age

There is an interesting discussion on the Association for Learning Technology mailing list: shoud student be allowed to use Kindle in open book examinations? It sterted with a question:

 I have been asked whether a student can bring her kindle into an open book exam, as she has e-copies of the relevant set texts. Logically I’d be worried about saying no, as it isn’t unreasonable for a student to buy set books in e versions. However, there are clearly problems in controlling what else might be on the kindle.”

Someone wrote:

Rather than consider how Kindles might affect the present system of open book exams (now about 40 years old) we might consider how Kindles (and, perhaps, Google) might affect how and what we examine. It is possible that an open book exam, under time pressures, might incorporate an internet search as part of its testing, with the requirement to reference findings as appropriate. Limiting the role of technology could be seen as rather like Canute.

In my opinion, the new technology makes obsolete the very concept of standardised and controlled assessment because anything which can be standardised can be much better done by a computer. For example, there is software — easily available on the Internet — which not just solves standard school and undergraduate level algebraic, trigonometric, logarithmic, differential equations, it produces a complete step-by-step verbal write-up of the solution, with a detailed explanation of every step; if the user wishes, more mundane symbolic rearrangements could be kept hidden, or, on the contrary, unrolled.

However, the solution to the challenges of new technology had been known for at least 70 years now, and used in educational establishments working at the cutting edge of technological progress of their time, say, in physics and mathematics university departments involved in training of researchers and engineers for the Manhattan project in the USA, and in similar establishments in Russia. This solution was an open book public oral examination, where a candidate was allowed (and actually encouraged) to consult books and records of his choice, while examiners reserved the right to ask any question — and many, if not most, statements were distinctively non-standard. Also, it was at examiners’ discretion to decide whether the candidate had to answer particular question on the spot or given a couple of hours for thinking. If the Internet existed in 1940-s, it would be instantly and painlessly adopted into the open book examination procedure.

Of course, such examinations are not allowed under the current legal framework which regulates British univerisites. But this is a sign of a real technological change: it does not necessarily fit in the existing legal set-ups.

 

Posted by: Alexandre Borovik | November 13, 2011

The Lancet on Bayesian statistics

The Lancet, of all journals, published a review of a popular book on Bayesian statistics.

With thanks to muriel.

Posted by: Alexandre Borovik | November 7, 2011

Herman Cain: “My mathematical training”

On page 6 of this NYT article, search for words “my mathematical training”.

Magazine Preview:  On the Ropes with Herman Cain
By T. A. FRANK
The end of the Cain campaign has been at hand for months. And yet the end doesn’t arrive. And the end isn’t about to arrive now either. 

With thanks to muriel.

Posted by: Alexandre Borovik | November 6, 2011

Amnesia Principle

A book picked at random off a shelf in the library of Mathematisches Forschungsinstitut Oberwolfach and opened at a random page revealed a little gem, a paper by Pierre Ageron Logic without self-deductibility, in: Logica Universalis (J.-Y. Beziau, ed.), Birkhauser, Basel – Boston -Berlin, 2007, pp. 87-93.

The “law of self-deductibility” is a simple principle “if A, then A” which expresses the reflexivity of entailment; apparently, its formulation can be traced back to Stoic philosophers. Pierre Ageron comments that

Self-deductibility has a paradoxical status. It seems so obvious that, unlike the law of exluded middle [...] it has hardly ever a matter of controversy [...]. Also it seems of no use whatsoerver in the mathematical practice and totally sterile in terms of deductive power.

However, then Pierre Ageron points out that Charles Pierce in hisAlgebra of Logic (1885)

mentions in passing quite a remarkable interpretation of the law of self-deductibility. Pierce referes to it as the “first icon of algebra” and argues that it “justifies our continuing to hold what we have held, though we may, for instance, forget how we were originally justified in holding it”. This clearly suggests that self-deductibility works as an amnesia principle: once some statement is proved, it allows us to forget how it was proved. In mathematical practice, it is obviously a good thing that we can continue to trust our old theorems even if we are unable to reconstruct their proofs.

In my book, I briefly discuss Timothy Gowers‘ observation that, in his opinion, a “comprehensible” proof is not necessarily the shortest one, but a proof of small width. Here, width measures how much you must hold in your head at any one time. Alternatively, imagine that you write a detailed proof on a blackboard, carefully referring to all intermediate steps.

However, if you know that a certain formula or lemma will never be used again, you erase it and re-use the space. A “small width” proof is a proof which never expands beyond one (small) blackboard. In other words, a “comprehensible” proof is a proof produced by a systematic application of the “amnesia principle” and the humblest of all tautologies, “if A then A”.

[Imported from my old and now defunct blog]

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