I suspect that the gap will only widen before it closes – as I believe it must.

Reason why I believe the gap must widen
==========================
As I see it, the focus of theoretical computer science (post-Turing) seems to be on discovering how humans can compute better; whilst the focus of mainstream mathematics (post-Goedel) seems to be on deciding how humans ought to reason better.

Thus, I would suspect that a post-Turing theoretical computer scientist would probably hold sub-liminally (based instinctively on experience with the reliability of the communication between mechanical artefacts) that SETI should be safe because nature is not malicious; and Turing’s analysis of computable numbers suggests that there must be a categorical language of effective, and unambiguous, communication between different forms of intelligence that would allow them to always co-operate despite having to compete for common resources.

However, I would suspect that a post-Goedelian mathematician would probably hold sub-liminally (based on an unquestioning faith in the validity of Goedel’s interpretation of his own formal reasoning in his seminal 1931 paper on formally undecidable arithmetical propositions) that SETI is dangerous because we should not presume that nature cannot be malicious; since Goedel’s conclusion of the existence of a formally undecidable arithmetical proposition suggests that there is no categorical language of effective, and unambiguous, communication between different species of intelligence that would allow them to always co-operate whilst competing for common resources.

See http://alixcomsi.com/25_Aristotlean_particularisation_Presentation_Update.pdf

If so, such sub-liminal beliefs should lead over time to increasingly differing motivations in deciding upon the amount of effort that one should put into developing results in a language (such as first order PA) that, prima facie, appears ideally suited for communicating effectively and unambiguosly (the ‘Can you understand my meaning?’ syndrome) by an objective yardstick (a finitary interpretation of PA in terms of Turing computability); as compared to the effort one shoulld put into developing results in a language such as ZF that, prima facie, appears ideally suited only for expressing oneself effectively and unambiguosly (the ‘I can understand my meaning’ syndrome) by an essentially subjective yardstick (belief that the axiom of infinity is ‘self-evidently true’, and so ZF must be consistent).

Reason why I believe the gap must close
==========================
Basing one’s efforts on the way things are will always be more productive than basing them on the way they ought to be.

To illustrate.

Apart from an almost ‘religious’ faith in the consistency of ZF, the faith that mainstream mathematics has in the validity of Goedel’s conclusions is founded essentially on the almost ‘religious’ belief that Aristotle’s particularisation ought to always hold over the natural numbers.

By this I mean the belief that, if I can show that it is not the case that a number-theoretic relation R(x) always holds over the natural numbers, then I may conclude that not-R(n) must hold for some natural number.

This belief is reflected in the introductory pages of any logic text (or paper), where it is almost ‘informally’ stated as almost ‘self-evident’ that the formula [(Ex)R(x)] of any formal first order logic / language (the mainstays of mathematics) may always be validly interpreted as ‘There exists some x such that R(x) holds’ over the domain of the interpretation.

However, I have yet to come across any literature that highlights the consequences (say, for Rosser’s incompleteness argument) of the fact that Aristotle’s particularisation holds under any sound interpretation of first-order Peano Arithmetic over the natural numbers if, and only if, PA is omega-consistent.

Now, Turing’s analysis of the Halting problem does not allow a theoretical computer scientist (as distinct from theoretical computer science – which uncritically accepts the mainstream mathematics of the day as the court of last appeal) to appeal to Aristotle’s particularisation when interpreting the discipline’s results to engineers for practical application (which, of course, is the very raison d’etre for most presentations in theoretical computer science).

Reason: A machine intelligence cannot conclude that if it is not the case that a number-theoretic relation R(x) always holds (is algorithmically computable as always ‘true’) over the natural numbers, then it may conclude that not-R(n) must hold (be computable as ‘true’) for some natural number

The distinction is seen even more vividly if we introduce adequate definitions that allow us to interpret the satisfiability of the atomic formulas of PA over the natural numbers:

(a) in terms of Turing-computability (as commonly understood by theoretical computer scientists);

(b) in terms of PA-provability (as commonly understood by mathematicians).

Now, it is not very difficult to construct such a set of definitions. The challenge is in recognising, first, that the ‘standard’ interpretation of PA – which appeals to Aristotle’s particularisation – is then a consequence of (b) and is inconsistent; and, second, that (a) yields a sound interpretation of PA which seems to imply that PA is categorical.

See http://alixcomsi.com/27_Resolving_PvNP_Update.pdf

http://player.bitgravity.com/debug/embedcode.php?ap=true&video=http%3A//bitcast-a.bitgravity.com/highbrow/livearchive40009/21aug-13.45to14.45.flv

Thank you very much for this journal from Hyderabad!

“Sir William Ostler once said ‘the transition from layman to physician
is the most awesome transition in the universe.’”
[Analog, July/Aug 2010, p.56]

If he said so, then the bootstrapping of my brain by the
neuromagnetometry/IR laser/entanglement/Quantum Advice download from the quantum computing ET was like transitioning me from a layman to a physician to the cosmos.

I had a bedside manner. I had a way of diagnosing anomalies in
physical systems, and a way of making prognosis on a partially
completed proof of a hitherto unknown truth, an unimagined truth, a
previously unimaginable truth.

Remember when Neo, played by Keanu Reeves in “The Matrix”, said:

“I know Kung Fu!”and then proved it, in simulated fight with Morpheus
played by Laurence Fishburne, asks:

“Do you think that’s air you’re breathing?”

Remember how it took more training before he could punch faster than
his muscles? Dodge bullets? Leap from rooftop to rooftop of
skyscrapers in Sydney, Australia? I recognized that
part of Sydney from a flyover by a drone on way to Tidbinbilla.

I know Konigsberg Bridges Fu!

I know Rubidium Bose-Einstein Condensate Fu!

I know Godel Incompleteness Bridges Fu!

I know Quantum Oracle Fu!

I know Stardrive Fu!

In which case, perhaps efforts to mitigate those professional gender imbalances in the last four decades would have been effective … which they have been in medicine, but by-and-large not in science, mathematics, and engineering.

Unless the University of Cambridge is surprisingly different from the University of Washington, the clinical conferences at Cambridge’s Addenbrooke’s Hospital will in general have no mathematicians attending, and correspondingly, there will be few or no physicians attending Cambridge’s mathematical conferences.

This is a pity, as this kind of cross-disciplinary experience is immensely illuminating! A well-written overview of the goals of a clinical conference is the British Medical Schools Council’s two-page Role of the Doctor Consensus Statement.

You will note in this consensus statement that ‘doctor’ explicitly encompasses all qualified doctors, including those in training. If there exists a similar consensus statement for mathematicians (British or otherwise), then I would be very interested to read it!

The proof of the educational pudding, though, is in the lecturing. Thus, if you were to arrange (via a colleague) to attend an Addenbrooke’s clinical conference—these conferences typically commonly are held in crowded rooms at shockingly early hours of the morning, but occasionally are held at jolly evening dinners; the former are more instructive—then I think you would have a very enjoyable and memorable time of it.

One thing you will come to appreciate is that there are some exceedingly important respects in which mathematical educational methods lead all other STEM disciplines, including medicine. Here in particular I have in mind ideals of naturality, which IMHO are defined better and taught more effectively in mathematics, than in any other discipline.

It is conceivable, therefore, that you might encounter physicians who would take a keen interest in mathematical methods of education.

A good mental model for an Artin stack is then the quotient of a manifold by a group which may no longer be finite, e.g. a Lie group. Thus the isotropy groups of points may be infinite, and may have non-trivial Lie group structure.

Medical grand rounds conferences are (in my experience) always tremendously interesting and clear; even more interesting and clear are morbidity-and-mortality conferences; for the simple reason that neither the speaker nor the audience patients to come to harm.

Thus medical talks are targeted—by a nearly inviolable tradition that began with William Osler—to that 1/3 of the audience who are *most* clueless. As for the remaining 2/3 of the audience, experience swiftly teaches them that they surely will encounter cases that challenge them, no matter how great their skills.

In consequence, at medical talks the whole audience listens with rapt attention.

Engineering talks generally are reasonably interesting and clear, because they are targeted to that 1/3 of the audience whose skills are moderate, and who want to improve those skills. Not to mention, the bottom 1/3 of an engineering audience wants to catch up, and the top 1/3 of the audience can reasonably hope to learn a few new tricks.

Thus at engineering talks (broadly speaking) the whole audience listens with moderate levels of attention.

Mathematical talks often are targeted to the top 1/3 of the audience who already grasp what’s going on. The middle 1/3 of the audience has to scramble to keep up … as for the bottom 1/3 of the audience .. well … they can always think about their own mathematical speciality.

Thus a pretty fair fraction of any given mathematical audience (again, broadly speaking) tends to tune-out of an advanced-level talk.

Arguably, all of this is as it should be. Because if medical talks were socially constructed like mathematical talks … that is, pitched only at the most knowledgeable members of the audience … then it would become unacceptably risky ever to go to a hospital, because there would be too many marginally competent physicians.

Conversely, if mathematical talks were socially constructed like medical talks … then the most advanced mathematics would never be transmitted at all … as a result, we would perhaps have more ordinary mathematicians … and perhaps fewer great ones.

Isn’t it true, that the mathematical community as a whole, would regard this as a bad trade-off? On the grounds that great mathematicians are more important to mathematics, than great physicians are to medicine?

Perhaps not too much change in this situation can be expected … so can we conclude that the present traditions are nearly optimal?

That would be my opinion … except that these traditions and trade-offs (IMHO) contribute greatly to the gender imbalance that is so markedly characteristic of mathematics, and which is so nearly absent (relatively speaking) in medicine.

That is why (IMHO) the question Tim asked is a deep one.

But then again it may not.