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ℓ(t)

@elloft.bsky.social

87 followers 23 following 1.8K posts

Philosophy, mathematics (logic), pedagogy, linguistics, psychology, aesthetics, data science, web dev, & poetry This is kind of just a notebook. Tags are for muting. Je comprends un peu le français 🇫🇷 Ich verstehe nur ein kleines bisschen Deutsch 🇩🇪

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ℓ(t) @elloft.bsky.social · Sep 20

I find myself thinking that certain degrees of both formalism and rhetoric are necessary for proper knowledge and understanding of certain topics—not one or the other. I then wonder whether this idea should be investigated by formal or rhetorical means, and I end up back where I started.

#philsky

ℓ(t) @elloft.bsky.social · 12h

p9 “We believe that the philosophy of mathematics should be consistent with scientific findings about the only mathematics that human beings know or can know.”

This aligns with our previous restriction of mathematics. Cf. chapters 2 and 3 (as of 10/19/2025).

ℓ(t) @elloft.bsky.social · 12h

“When the full metaphorical character of mathematical concepts is revealed, such confusions and apparent paradoxes disappear.”

(A pedagogical note)

ℓ(t) @elloft.bsky.social · 12h

Ibid “Many of the confusions, enigmas, and seeming paradoxes of mathematics arise because conceptual metaphors that are part of mathematics are not recognized as metaphors but are taken as literal.

ℓ(t) @elloft.bsky.social · 12h

“To do so, it must reveal how mathematics is grounded in embodied experience and how conceptual metaphors structure mathematical ideas.”

N.B. “experience”

ℓ(t) @elloft.bsky.social · 12h

“We believe that cognitive science can, in many cases, dispel the paradoxes and clear away the shrouds of mystery to reveal in full clarity the magnificence of those ideas.

ℓ(t) @elloft.bsky.social · 12h

“Up to now, even the basic ideas of college mathematics have appeared impenetrable, mysterious, and paradoxical to many well-educated people who have approached them.

ℓ(t) @elloft.bsky.social · 12h

Ibid “Yet many of [math’s] beauties and profundities have been inaccessible to nonmathematicians, because most of the cognitive structure of mathematics has gone undescribed.

ℓ(t) @elloft.bsky.social · 12h

“In those days, thought was taken to be the manipulation of purely abstract symbols and all concepts were seen as literal—free of all biological constraints and of discoveries about the brain. Thought, then, was taken by many to be a form of symbolic logic.”

Here, I’m reminded of Wittgenstein.

ℓ(t) @elloft.bsky.social · 12h

“Insights of the sort we will be giving throughout this book were not even imaginable in the days of the old cognitive science of the disembodied mind, developed in the 1960s and early 1970s.

ℓ(t) @elloft.bsky.social · 12h

p5 “As will become clear, it is only with these recent advances in cognitive science that a deep and grounded mathematical idea analysis becomes possible.

ℓ(t) @elloft.bsky.social · 13h

“Therefore, human mathematics which is constituted in significant part by conceptual metaphor) cannot be a part of Platonic mathematics, which—if it existed—would be purely literal.”

ℓ(t) @elloft.bsky.social · 13h

p4 “The argument [against Platonic mathematics] rests on analyses we will give throughout this book to the effect that human mathematics makes fundamental use of conceptual metaphor in characterizing mathematical concepts. Conceptual metaphor is limited to the minds of living beings.

ℓ(t) @elloft.bsky.social · 13h

Speaking of which, I wonder if they’ll end up implicitly making a case for structuralism.

ℓ(t) @elloft.bsky.social · 13h

I take it that they’re not Platonists then. Thank God.

(Pun intended.)

ℓ(t) @elloft.bsky.social · 13h

“Human mathematics, the only kind of mathematics that human beings know, cannot be a subspecies of an abstract, transcendent mathematics.”

ℓ(t) @elloft.bsky.social · 13h

pXVI “But the more we have applied what we know about cognitive science to understand the cognitive structure of mathematics, the more it has become clear that this romance cannot be true.

ℓ(t) @elloft.bsky.social · 13h

Notes on Where Mathematics Comes From by #Lakoff and Nuñez

#mathsky #cogsci #philsky

$The Romance of Mathematics In the course of our research, we ran up against a mythology that stood in the way of developing an adequate cognitive science of mathematics. It is a kind of "romance" of mathematics, a mythology that goes something like this. • Mathematics is abstract and disembodied-yet it is real. • Mathematics has an objective existence, providing structure to this universe and any possible universe, independent of and transcending the existence of human beings or any beings at all. • Human mathematics is just a part of abstract, transcendent mathematics. • Hence, mathematical proof allows us to discover transcendent truths of the universe. • Mathematics is part of the physical universe and provides rational structure to it. There are Fibonacci series in flowers, logarithmic spirals in snails, fractals in mountain ranges, parabolas in home runs, and π in the spherical shape of stars and planets and bubbles. • Mathematics even characterizes logic, and hence structures reason it-self-any form of reason by any possible being. • To learn mathematics is therefore to learn the language of nature, a mode of thought that would have to be shared by any highly intelligent beings anywhere in the universe. • Because mathematics is disembodied and reason is a form of mathematical logic, reason itself is disembodied. Hence, machines can, in principle, think.$

ℓ(t) @elloft.bsky.social · 2d

I suppose this is as opposed to deontological aesthetics, where the application of rules statically determines when a deduction is beautiful.

ℓ(t) @elloft.bsky.social · 2d

Cf. Mathematical Beauty §7.2 as of 10-17-2025.

\item A paragraph is beautiful with respect to the overarching understanding it induces in the reader. (teleological aesthetics)

ℓ(t) @elloft.bsky.social · 2d

We thus arrive at a bijective correspondence of terms:
sentence \bic sentence
paragraph \bic paragraph
essay \bic essay,
and, if we really want to push it,
treatise \bic treatise

ℓ(t) @elloft.bsky.social · 2d

But there seems to be lots of structure that would be lost if we were to flatten an essay or sequence of proofs into one long paragraph.

ℓ(t) @elloft.bsky.social · 2d

It’s just that rhetorical paragraphs are in fact tuples of sentences, but they don’t seem to necessarily be deductions. But, on the other hand, both rhetorical and logical paragraphs are evidential.

ℓ(t) @elloft.bsky.social · 5d

But I suppose from an execution standpoint, the original ternary analogy works; it may just depend on which relation is on display in the analogy.

ℓ(t) @elloft.bsky.social · 5d

Furthermore, it hasn’t occurred to me until now that, because deductions are actually paragraphs, which essays are made of, a formal analogue to an essay might be something consisting of multiple proofs, rather than just one.

ℓ(t) @elloft.bsky.social · 5d

On a pointless quest to get rid of all two-word discipline names.

atoms : analysis :: molecules : synthesis

Words & sentences : analysis :: paragraphs & essays : synthesis

Also, I really need to write a LaTeX command for analogies.

#philsky