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I’ve been reading Zoltán Kövecses’s Extended Conceptual #MetaphorTheory. I was going to do a notes thread, but I’m hesitant bc I feel it would just be a steady stream of complaints and disagreements. Perhaps I should just move on to something else. hmmm

#linguistics #cognitivelinguistics #cogsci

#hottake: The quickest way to learn one math skill is through drills of that skill, but the quickest way to build overall math competence is through problems that draw on any number of skills at once, the same way reading builds general vocabulary faster than vocabulary lists. #MathSky

An infinite green coordinate grid seen in perspective against a black background. The standard parabola is plotted in blue, and integer points are marked with gold dots. From our vantage point, the parabola is circular, with two limbs stretching toward the horizon at infinity. The equation of the parabola floats at this vanishing point on the y-axis.

Working on a set of mathematical postcards, both digital "scenes" and overlays on photos. Here we see the parabola y=x^2, a conic section, as a circle.

For now this is a 3-inch matte black self-adhesive sticker:

diffgeom.com/products/par...

#MathSky #MathArt #SciArt

Skinny Homer Simpson meme. Top skinny Homer labelled "550 Contributions in the last year" (referring to Github contributions). Bottom Homer with his fat pulled back behind him labelled "Private repo of philosophy notes and poetry written in LaTeX". Top and bottom Marge Simpson labelled "Linkedin".

I'm alright just been studying bc I am a nerd

a chain-like directed graph that ends with a node "T is S" pointing to two nodes T = S and T ≈ S.

This took a long time to make and idk what to do now.

This case against formalism appears to also be contrary to this old idea of mine and to any other fixations related to data engineering.

bsky.app/profile/varn...

p5 “\dots{}informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.”

“OMEGA: And what happens if even your 'sophisticated conjectures' are falsified and if this time you cannot replace them by unfalsified ones?”

Falsifiability? Popper?

p13 “TEACHER: 'Plausible' or even 'trivially true' propositions are usually soon refuted: sophisticated, implausible conjectures, matured in criticism, might hit on the truth.

[italics added]

Here, his philosophy of science shines through.

It is certainly not a common thought that a mathematician might need to perform \textit{scientific} induction in order to successfully write a proof.

“Both the Editor and Euler use for this process of discovery the modern term '\textit{induction}' instead of the ancient analysis' (ibid.). The \textit{heuristic} precedence of the result over the argument, of the theorem over the proof, has deep roots in mathematical folklore. Let us quote…”

Ibid “We read in the editorial summary to Euler [1756-7] that arithmetical theorems 'were discovered long before their truth has been confirmed by rigid demonstrations.

Ibid “That conjectures (or theorems) precede proofs in the \textit{heuristic} order was a commonplace for ancient mathematicians.” [italics added]

Do we find at long last that heuresis itself (or some aspects of it) are the methodological key to analytical pursuits?

A distant body of knowledge…a folk theory?

p10 “Till then I propose to retain the time-honoured technical term 'proof for a thought experiment - or 'quasi-experiment' - which suggests a decomposition of the original conjecture into subconjectures or lemmas, thus embedding it in a possibly quite distant body of knowledge.”

Case in point:
p4 “The history of mathematics and the logic of mathematical discovery, i.e. the \textit{phylogenesis} and the \textit{ontogenesis} of mathematical thought, cannot be developed without the criticism and ultimate rejection of formalism.” [italics added]

…I should not be surprised if some enthusiastic pupil of Vesalius in those glorious days of early anatomy, when the powerful new method of dissection emerged, had identified biology with the analysis of dead bodies.”

But few will infer from this that human beings are 'suitable for scientific investigation' only when they are 'presented in "dead" form', and that biological investigations are confined in consequence to the discussion of dead human beings - although,…

Case in point:
p3 “Nobody will doubt that some problems about a mathematical theory can only be approached after it has been formalised, just as some problems about human beings (say concerning their anatomy) can only be approached after their death.

Haven’t even made it past the introduction of Proofs & Refutations, and it’s already much more metaphorically rich, sensible and interesting than Lakoff’s WMCF.

I suppose if I have any further notes or musings on P&R, I’ll put them here.

#mathsky #mathematics #philsky #philosophy #Lakatos

Tempted to release another LaTeX package on CTAN, but what I’ve currently got is suboptimal, and getting it ready to be published would probably be a lot of work…

hmmm

Here, I’m reminded of my game theory professor from university and his insistence that math majors take the course.

“If this is correct, then what the social sciences need to do to catch up is to ape the methods of the natural sciences. To some extent this has already happened. The increasing use of mathematics in the social sciences may be partly a result of this attitude.”

“Why should this be so? It can hardly be because natural scientists are smarter than social scientists. One possible answer is that the methods of the natural sciences are superior.

p116 “It is often felt that natural sciences such as physics, chemistry, and biology are in a more advanced state than social sciences such as economics, sociology, and anthropology; the former can formulate precise laws with great predictive power, while the latter usually cannot.

Would we shy away from a scenario where the role of philosophy is to speculate about uncertainties and unknowns?

“Moreover, the idea that philosophical enquiry and scientific enquiry are autonomous, each relying on their own methods, has been criticized as wishful thinking; opponents point out that while there is certainly progress in science, progress in philosophy is rather harder to discern.”

“Indeed the pool of questions classed as ‘philosophical’ has shrunk over the centuries, as more and more get appropriated by science.

“However, against this, many of the questions discussed in the history of philosophy, e.g. about perception, imagination, and memory, have turned out to be matters for the empirical sciences, in particular psychology.