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At the beginning of the last century, the Italian mathematician Giuseppe Peano published a work (the Formulario Mathematico) which expresses in symbolic language a number of definitions and theorems to which demonstrations are attached. More precisely it starts with elementary logic, continues with algebra results and ends with differential calculus.

factorial

As an example, after the definition of the factorial, Peano immediately states a result due to Pascal.

What I like in this kind of book is the purity of mathematics in the sense that no exercise or example disturbs the content. Ideally, it should cover the broad outlines of the major fields of mathematics.

To my knowledge, no work of this type has been done since then except for a few result booklets written by Bourbaki as an introduction to the areas of mathematics they wanted to develop. Is this really the case?

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    $\begingroup$ What exactly do you mean by formulary? $\endgroup$ – mrtaurho Feb 4 at 20:49
  • $\begingroup$ A document presenting in an organized way formulas, thus in our case theorems. $\endgroup$ – user601568 Feb 4 at 20:54
  • $\begingroup$ To what extent? Whole of mathematics known today; unlikely. Even for a single field this is nearly impossible to accomplish. The closest you could get is to look in a standard references of particular fields. $\endgroup$ – mrtaurho Feb 4 at 20:58
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    $\begingroup$ So in less fancy language, you're looking for texts which just give lots of (definitions and) theorems (with or without proofs included?) - ideally covering the "fundamental" results of some of the "main" topics of mathematics. Is that accurate? $\endgroup$ – Noah Schweber Feb 4 at 21:15
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    $\begingroup$ If you are looking for something which starts from pure logic and tries to build up mathematics from there, perhaps you would be interested in reading recent formalization attempts? There's been a big push lately to translate math into a language computers can understand, which is pure logical syntax. Reading the code is more involved than reading a book, but it's a contemporary project that is possibly related to your interests. Lots of theorems have been formalized in this way, but there's still a lot of work to do $\endgroup$ – HallaSurvivor Feb 6 at 20:52
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Since no one has said it, I'm going to give the obvious answer:

The only other "formulary" that I know of is Whitehead and Russell's Principia Mathematica. In some ways it is a successor to Peano's work, though it focuses primarily on logic and developing the foundations of mathematics. Part of the reason for this is that proofs of more complex topics, like analysis, become absurdly long when written Principia's symbolic language.

One other work that comes to mind which you might be interested in is Frege's Begriffsschrift. It predates Peano's work considerably, but makes extensive use of the symbols-definitions-theorems format you describe.

I would also recommend looking at the Metamath proof database, which includes thousands of formal proofs derived from basic logical axioms.

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