@Sebastian
PS
unbedingte Leseempfehlung: Sydney Padua, The Thrilling Adventures of Lovelace and Babbage: The (Mostly) True Story of the First Computer

Aus dem Buch ist der Cartoon oben.
#boolean #boole #tea

Alicia #Boole au pays des #polytopes :

Au départ, il y a les cinq « solides platoniciens » vénérés en géométrie depuis l'Antiquité : le cube, le tétraèdre, l’octaèdre, le dodécaèdre et l’icosaèdre. Mais pourquoi s’arrêter aux 3 dimensions de l’espace ordinaire ? Alicia Boole Stott a consacré sa vie à chercher des solides réguliers en dimension 4… et elle a trouvé !

Une inspiration pour la future itération des #polyharmonies ;-)

https://www.arte.tv/fr/videos/107398-006-A/voyages-au-pays-des-maths/

> Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.
https://gutenberg.org/ebooks/15114
#Boole #GeorgeBoole #StatisticsClass #GutenbergEbook

In 1847, George Boole developed Boolean algebra, on which all digital computers are based. #Poetry #Science #History #Logic #Boole (https://sharpgiving.com/thebookofscience/items/p1847b.html)

Overall thoughts: this is such a shockingly original work that no wonder it caught on slowly, and we can certainly forgive all errors and infelicities of presentation. #WSJevons (mentioned above) led one response by acknowledging #Boole 's insights but trying to fold them into the old paradigm. But with developments such as emphasing inference over equality, abandoning partially defined connectives, and new quantifiers to allow binary+ relations, the formal approach to #logic was unstoppable.

#Boole 's closing chapter, with untranslated quotes in French, Italian, and ancient Greek, was not an easy read, but he compares and contrasts the abstraction from physical observation to mathematics with his abstraction from thought to mathematical #logic , noting that correctness is a criteria that can be meaningfully applied to thought processes but not to physical ones; and observes that just as physical science cannot be entirely reduced to maths, so it is with the science of the intellect.

#FinishedReading #Boole 's Laws of Thought, whose final chapters partly shift attention from #logic to #probabilityTheory and #philosophyOfScience . I don't have a strong sense of his historical standing in either discipline, although for probability there is https://en.m.wikipedia.org/wiki/Boole%27s_inequality . The attraction of probability is clear, with its range of values from 0 to 1, use of '1 -' for negation, multiplication for conjunction (of independent events) etc. There are rhymes here with his logic at least

Oof. #Boole #logic

The section where #Boole compares his #logic with Aristotelian syllogisms, the dominant approach in the West for more than two millennia, would have been key for readers of the time. I imagine the attached quote would have been unbelievably spicy, but virtually nobody would disagree with it now. With the help of some tidying up and extensions from Peirce, Lewis, Schroeder, Frege etc, Boole's vision won comprehensively, and helped to build our modern world.

On page 170 we finally see the conditional, if-then. This is in a section on 'secondary propositions' which relate the truth of propositions. If y then x is not, perhaps surprisingly given #Boole 's mission to arithmetise #logic , encoded as the exponent xʸ, but instead via introduction of a new unknown v, as y=vx (v and x). Given that conjunction as multiplication led us to elimination via semantically dubious propositional division, I suppose I should feel lucky we avoided logical logarithms!

To modern eyes a big thing missing in #Boole 's #logic is any proof of soundness, let alone completeness; he has what we would call a semantics (propositions as subsets of all entities in the universe of discourse, connectives as set operations e.g. conjunction as intersection) and many formula manipulations, but no verification of the manipulations in general. The approach is justified in Chapter 1 by " the general truths of Logic... when presented to the mind... at once command assent".

A more twisted, but important, example: say x=yz, e.g. slithy = lithe and slimy. What can we say about y (lithe)? We rearrange as y=x/z, but division has no logical interpretation, so we develop the right hand side to xz + (1/0)x(1-z) + (0/0)(1-x)(1-z), with the (1-x)z case disappearing under coefficient 0. Conclusion: the lithe things are all the slithy slimy things, plus no, some, or all of the non-slithy non-slimy things. Independently, nothing is both slithy and not-slimy. #Boole #logic

What if the coefficients, e.g. f(x,y) for x,y ranging over 0 and 1, are not themselves 0 or 1? #Boole 's specific answer for case 0/0 is that the truth value is indeterminate; for any other value, including 1/0, we get the independent conclusion that the part of the developed proposition that the coefficient is applied to is 0. For example, x+y develops to 2xy + x + y, which is x + y with independent conclusion that the conjunction xy is 0 (false) - in keeping with + as disjoint union #logic

#Boole sees #logic as arithmetic specialised to 0 and 1, as any function can be 'developed' e.g. by sending f(x,y) to f(1,1)xy + f(1,0)x(1-y) + f(0,1)(1-x)y + f(0,0)(1-x)(1-y). This looks like truth tables, considering all combinations of true and false, but this development only needs to happen at the end of a chain of reasoning, so intermediate terms do not have to be logically interpretable, by analogy with the use of i in finding real roots of polynomials. This raises soundness questions!

The treatment of quantification by #Boole is interesting (quantifiers came along 20ish years later, with Frege). 'All x are (have property) y' is translated as x = vy (read concatenation as conjunction) where v is a distinguished 'indefinite' proposition, except that it must contain at least one y, understood to indicate that it is possible to find a subset of y that matches the collection x. 'Some x are y' is translated as vx = vy, where v must contain at least one each of x and y. #logic

The more I think about it, the more baffling I find #Boole 's insistence that x and x and x cannot be said to equal x (in his notation, x³ = x). One would think that two applications of the axiom x² = x (and replacing equals by equals) would get him there. It's like he confused the failure of a proof with the failure of the theorem. #logic

This footnote is an example of the arithmetical approach to #logic making life terrible for #Boole ; given that x² (i.e. x and x) = x is an axiom, shouldn't x³ = x hold? Apparently not, as x³ - x = 0 'factorises' into gibberish terms like 1 + x (we can't add new things to the universe), or -1, which has no meaning at all (not to be confused with the negation of 1, which is 1 - 1 = 0). I must admit to my doubts about the well-definedness of this whole enterprise!

#Boole develops #logic by close analogy with arithmetic, though he is at pains to say this is mere analogy and there is no a priori reason the rules should be the same. So while we usually think of logic as being about entailment, Boole virtually ignores it in the early going and makes equality primary; see the attached proof of the principle of contradiction (here 1 stands for the whole universe, and x - y, defined only if y is a subset of x, is set difference), with its arithmetical flavour.

#Boole 's propositions do not range merely across 0 and 1, as often presented today, but across subsets of all objects in the universe (or some agreed upon universe of discourse). If this sounds like Boolean Algebra, you're half right; conjunction is indeed intersection, but disjunction (which he writes +) is *disjoint* union, so x+y is not meaningfully defined in general, as with x/y in arithmetic (as y might be 0). This strikes me as something which might cause trouble later. #logic

#AmReading this 1854 book by George #Boole , which summarises his thoughts (first published a few years earlier) on #logic , as well as probability. Boole built the world I live in as a logician (and to extent, the world we all live in in the age of computers) but this is the first time I've read him in the original, so I thought I might make a thread with a few notes in it as I read it over the next few weeks.