### mathematics and statistics online

You ask about where, eventually, the book should go—I would prefer it to go to someone or some where wh. Still, as I sat at breakfast, I thought it only proper that I should look through the book page by page.

The book was published in , yet its clean formalism and, yes, despite the language barrier, I could read the math is pretty much as one would write it today. On page 9 I started to see underlinings and little marginal notes, all written in light pencil. I kept on flipping pages. After a few chapters, the annotations disappeared. But then, suddenly, tucked into page , there was a note:.

It was in German, with what looked like fairly typical older German handwriting. And it seemed to have something to do with Lagrangian mechanics. I kept flipping through the book. No more annotations. But then, on page , a bookmark—with a charmingly direct branding message:.

Would there be anything more? I continued flipping. Then, near the end of the book, on page , in a section on the relativistic theory of electrons, I found this:. But what on Earth was it doing here? Remember, the book is about quantum mechanics. Quintessential Turing stuff. So, I immediately wondered, did Turing write this page? And soon I had to go, carefully packing the book away, ready to pursue the mystery of what this page was, and who wrote it.

The preface by Dirac is dated May 29, ; the one from the translator— Werner Bloch —August 15, Why did Alan Turing get the book in German rather than English? But in those days, German was the leading language of science, and we know Alan Turing knew how to read it. Did Alan Turing buy the book, or was he given it?

Could this have been a price in Reichsmarks , suggesting the book was sold in Germany? Alan Turing was born June 23, coincidentally, exactly 76 years before Mathematica 1. He got his undergraduate degree after the usual three years, in In the s and early s, quantum mechanics was hot, and Alan Turing was interested in it.

We also know that in , he asked the Cambridge physicist Ralph Fowler for a possible question to study in quantum mechanics. Fowler suggested computing the dielectric constant of water —which actually turns out to be a very hard problem, basically requiring full-fledged interacting quantum field theory analysis, and still not completely solved. Given that there seems to be a used price in the book, Turing presumably bought it used. Who was its first owner? The annotations in the book seem to be concerned primarily with logical structure, noting what should be considered an axiom, what logically depends on what, and so on.

What about the note tucked into page ? But what does the note say? Was the note perhaps written by an editor of the journal? So, was this note written by Max Born?

- Collected Works of A.M. Turing : Mathematical Logic (Turing, Alan Mathison, Works.).
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The marketing copy is quaint and rather charming. But when is it from? But for more than 70 years ending in Heffers was located, as the bookmark indicates, at 3 and 4 Petty Cury. But, OK, so the bookmark was from before But how much before? But for our purposes it seems as if we can reasonably conclude that our book came from Heffers which was the main bookstore in Cambridge, by the way sometime between and OK, so we know something about when the book was bought. When was it written? Well, of course, lambda calculus had to have been invented.

And that was done by Alonzo Church , a mathematician at Princeton , in an initial form in , and in final form in To fill in a little more of the timeline: from September to July with a break of three months in the summer of , Turing was at Princeton, having gone there to be, at least nominally, a graduate student of Alonzo Church. Turing was back in Cambridge in July , but already by September of that year he was working part-time for the Government Code and Cypher School —and a year later he moved to Bletchley Park to work full time on cryptanalysis.

After the war ended in , Turing moved to London to work at the National Physical Laboratory on producing a design for a computer. He spent the —8 academic year back in Cambridge, but then moved to Manchester to work on building a computer there. In , he began working in earnest on theoretical biology. But all this came to an end on June 7, , when Turing suddenly died.

- SMALL UNIT ACTIONS (Part 2 of 2).
- A Book from Alan Turing… and a Mysterious Piece of Paper.
- L'actualité de Turing : l'inférence dans la pensée, le calcul et les formes biologiques.

OK, but back to the lambda calculus page. But can we date the paper? One writes f [ x ] to mean a function f applied to an argument x. And there are lots of named functions that f can be—like Abs or Sin or Blur. But is there still something we can write for f that will make f [ x ] be this? In traditional mathematics, functions tend to be thought of as things that map inputs like, say, integers to outputs that are also, say, integers.

See the rather obscure-looking line PI1IIx? What does it mean? Ordinary composition of functions is pretty familiar from mathematics. But does one really need the brackets? In the Wolfram Language f g x is an alternative notation. But what would f g x mean? And if f and g were ordinary functions in mathematics, this would basically be meaningless. But if f is a higher-order function , then f[g] can itself be a function, which can perfectly well be applied to x.

In f[x] the f is a function of one argument.

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## Alan Turing - Wikiquote

And f[x] is equivalent to Function[a, f[a]][x]. But what about a function of two arguments, say f[x, y]? What would this be? And such objects are called combinators. Combinators have a long history. Applied ethics.

### Volume 4. Mathematical Logic

History of Western Philosophy. Normative ethics. Philosophy of biology. Philosophy of language. Philosophy of mind. Despite this, he was successfully entered for Sherborne School. The headmaster soon reported, almost correctly: 'If he is to be solely a scientific specialist, he is wasting his time at a Public School' Hodges , Turing's private notes on the theory of relativity showed a degree-level appreciation, yet he was almost prevented from taking the school certificate lest he shame the school.

## Turing, Alan 1912-1954

The stimulus for communication and competition came only from another very able pupil at Sherborne, Christopher Morcom , to whom he found himself powerfully attracted. Morcom gave Turing a vital period of intellectual companionship, which ended with the former's sudden death in Turing's conviction that he must now do what Morcom could not apparently sustained him through a long crisis. His thoughts turned to the question of how the human mind was embodied in matter, and whether, accordingly, it could be released from matter by death. This led him to wonder whether quantum mechanical theory affected the traditional questions of mind and matter.

As an undergraduate at King's College, Cambridge, from , he entered a world more encouraging to free-ranging thought. His reading of J. Turing's homosexuality became a definitive part of his identity, and the special ambience of King's College gave him a first real home.