Corpus evidence — top 10 passages

Most-relevant passages from the entire indexed corpus (67,286 paragraph chunks across YouTube transcripts, PubMed, arXiv, archive.org, Stanford Encyclopedia of Philosophy, OpenAlex, and more) ranked by semantic similarity (bge-small-en-v1.5).

1. 01 · blog0.744

   This second part, dwarfed by the properly mathematical one, can be called the pre-mathematical part, while I might refer to the other as the rigid part , since it forms a structure characterized by the absolute rigor of its inference rules . Though it is tempting to ignore how rigorous these rules are, that does not make them any less rigorous. The rigid part is, as I said, mathematics proper. It proceeds by mathematical proof and mathematical definition. (Pasch 1918 [2010: 51], our emphasis) Pasch thus distinguished between mathematics and what we may call “pre-mathematics”. Mathematics is en…

   blog/plato-stanford-edu/deductivism-in-the-philosophy-of-mathematics.md

2. 02 · blog0.740

   For example, to demonstrate Pythagoras’s theorem—that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides—we may assume as ‘given’ a right-angled triangle with the three squares drawn on its sides. In investigating the properties of this complex figure we may draw further (auxiliary) lines between particular points and find that there are a number of congruent triangles, from which we can begin to work out the relationship between the relevant areas. Pythagoras’s theorem thus depends on theorems about congruent triangles, and once t…

   blog/plato-stanford-edu/analysis.md

3. 03 · blog0.738

   The structure that results is a kind of laminated structure, a metaphysical “onion” with several layers. On this picture, of course, substantial and accidental forms are both “layers of the onion” in exactly the same sense. The distinction between essential and accidental features of a thing would therefore have to be drawn in some other way. If this reconstruction is more or less correct, then it is clear why universal hylomorphism and plurality of forms can be viewed as conceptually linked. Both fit nicely with the view that the structure of reality is accurately mirrored in true predication…

   blog/plato-stanford-edu/binarium-famosissimum.md

4. 04 · blog0.737

   The strength and accuracy of the versions of the argument these thinkers provide depend, partially at least, on the precision of their construal of the notion of equality of geometric magnitudes. It has been shown that some of Muslim thinkers have pretty detailed accounts of this notion (R. Rashed 2019). Like the other two arguments, the primary goal of the Mapping Argument is to show that no infinite continuous magnitude actually exists. Having read Euclid’s Elements (bks 7–9), Muslim thinkers knew that numbers could be easily represented by magnitudes. Therefore, any argument for the impossi…

   blog/plato-stanford-edu/arabic-and-islamic-philosophy-of-mathematics.md

5. 05 · blog0.736

   The conception is exemplified above all in such texts as Euclid’s Elements and such works of architecture as the Parthenon, and, again, by the Canon of the sculptor Polykleitos (late fifth/early fourth century BCE). The Canon was not only a statue deigned to display perfect proportion, but a now-lost treatise on beauty. The physician Galen characterizes the text as specifying, for example, the proportions of “the finger to the finger, and of all the fingers to the metacarpus, and the wrist, and of all these to the forearm, and of the forearm to the arm, in fact of everything to everything…. Fo…

   blog/plato-stanford-edu/beauty.md

6. 06 · blog0.735

   Note that, according to Avicenna, the claim that a separate mathematical object, say a triangle, exists cannot be justified unless we have come to know it through knowing a sensible counterpart of it that exists in the material world. Now if that separate triangle is the cause of any material thing, it must in the first place be the principle of its own sensible counterpart, or so Avicenna believes. But if the sensible triangle is caused by the separate triangle, then we can legitimately ask why the former needs the latter. It is either the essence or (some of) the accidents of the sensible tr…

   blog/plato-stanford-edu/arabic-and-islamic-philosophy-of-mathematics.md

7. 07 · blog0.735

   Now audible proportion may be no more than an accidental unity, but it is still the case, Grosseteste holds, that proportion is a necessary formal part of the real nature of harmony, just as its audibility is. So the subject of music, harmony, is, as it were, proportion realized in a certain matter, just as the actual material constitution of a natural object is the realization of some higher form, say that of an animal, the nature of whose functioning can be understood quite independently of the realization of it in that particular matter. Grosseteste sees subalternation as a phenomenon revea…

   blog/plato-stanford-edu/medieval-theories-of-demonstration.md

8. 08 · blog0.734

   Thus, it seems that there is no convincing justification for why a separate mathematical object must be the cause of its sensible counterpart, let alone the cause (or principle) of any other natural thing. Avicenna takes this argument as refuting (PM) . These arguments show that mathematical objects are neither separate entities fully detached from the sensible world nor the causes of natural things. Avicenna’s refutation of Platonism and Pythagoreanism regarding mathematical objects was so convincing and influential that these approaches almost completely disappeared in post-Avicennian philos…

   blog/plato-stanford-edu/arabic-and-islamic-philosophy-of-mathematics.md

9. 09 · blog0.733

   The difficulty is that the argument is not about precision. It concerns an objection to Aristotle that man qua man is indivisible, but geometry studies man qua divisible. Since man is not divisible, the principle of qua -realism, that if one can study X qua Y then X is Y , is violated. Aristotle says that man is in actuality indivisible (you cannot slice a man in two and still have man or men), but is materially divisible. It is enough that X is Y materially or in actuality to study X qua Y . Nonetheless, the solution to the puzzle could point to an Aristotelian solution to the problem of prec…

   blog/plato-stanford-edu/aristotle-and-mathematics.md

10. 10 · blog0.731

    Mesolabe A similar approach is taken by Descartes when he treats the problem of constructing mean proportionals, where in this case, he appeals to his famous mesolabe compass, an instrument that is used in Book Three of the La Géométrie to solve the same problem. As in 1637, this compass is used to construct curves (the dotted lines in figure 3 ) that allow us to identify the mean proportionals between any number of given line segments. And as Viète before him, in the Private Reflections Descartes uses this construction of mean proportionals to identify the roots of standard-form cubic equatio…

    blog/plato-stanford-edu/descartes-mathematics.md