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 · blog0.767

   Exploring this philosophical ground is indeed fruitful; for a careful and illuminating discussion that does this, see the entry The Epistemology of Visual Thinking in Mathematics . Nevertheless, understanding diagrams is not a philosophical dead end. In this section we survey the philosophical work that approaches diagrams in mathematics from this perspective, where what is examined are not individual diagrams in relation to individual mathematical propositions or proofs but rather diagrammatic proof methods in relation to mathematical subjects. The most prominent case of a diagrammatic proof …

   blog/plato-stanford-edu/diagrams-and-diagrammatical-reasoning.md

2. 02 · blog0.762

   There is thus in De Toffoli’s account an important connection between the proofs of a diagrammatic method and the proofs corresponding to them in a sentential axiomatization. The existence of the latter underlies our identification of the former as proofs. De Toffoli stresses that this does not mean that there are no significant epistemological advantages of the former over the latter. The differences between them however are not for her to be understood as logical differences as they are for Larvor. In his alternative picture, diagrammatic methods are freestanding. They have their own topic s…

   blog/plato-stanford-edu/diagrams-and-diagrammatical-reasoning.md

3. 03 · archive0.762

   reader has to object to the deduction of Archimedes. This deduction from simple and almost self-evident theorems may charm a mathematician who either has an affection for Euclid's method, or who puts himself into the appropriate mood. But in other moods and with other aims we have all the reason in the world to distinguish in value between getting from one proposition to another and conviction, and between surprise and insight. If the reader has derived some usefulness out of this discussion, I am not very particular about maintaining every word I have used.

   archive/sciemechacritica00machrich/sciemechacritica00machrich_djvu.txt

4. 04 · blog0.755

   Starting with some simple infinity constructed initially—it doesn’t matter which one we start with, given their isomorphism—he “creates” new objects corresponding to its elements, thereby introducing a distinguished simple infinity Dedekind calls “the natural numbers”. As we saw, this last step is parallel to his introduction of “the real numbers” (a parallel confirmed in Dedekind 1888b); but some aspects are clearer in the present case. Namely, the newly created objects are characterized completely by all arithmetic truths, i.e., by those truths transferable, or invariant, in the sense explai…

   blog/plato-stanford-edu/dedekind-s-contributions-to-the-foundations-of-mathematics.md

5. 05 · blog0.751

   (The order is unimportant: one can go through the derivation first and then follow the soundness proof.) That entire process would constitute thinking through a proof of the conclusion; and the diagrammatic thinking involved would not be superfluous. Shin et al. (2013) report that formal diagrammatic systems of logic and geometry have been proven to be sound. People have indeed followed proofs in these systems. That is enough to refute claim (a), the claim that all diagrammatic thinking in thinking through a proof is superfluous. For a concrete example, Figure 1 presents a derivation of Euclid…

   blog/plato-stanford-edu/the-epistemology-of-visual-thinking-in-mathematics.md

6. 06 · blog0.748

   (For an entry point on literature concerning the Third Man Argument, as well as the dialogue more broadly see the entry on Plato’s Parmenides ). Similarly, there has been much disagreement about which of the theses that underlie the TMA Plato was willing to give up in order to avoid the regress. To a modern reader the “self-partaking” premise seems like an obvious candidate, since it is quite odd to think that the form of largeness is itself large, the form of smallness itself small, etc. Plato’s TMA featured prominently in Aristotle’s Peri Ideon and was later picked up and discussed by mediev…

   blog/plato-stanford-edu/bradley-s-regress.md

7. 07 · blog0.748

   Philosophers standardly refer to sentences of the first set as “synthetic,” those of the second as (at least apparently) “analytic.” (Members of set III. are sometimes said to be “analytically false,” although this term is rarely used, and “analytic” is standardly confined to sentences that are regarded as true.) We might call sentences such as (5)-(10) part of the “analytic data” to which philosophers and linguists have often appealed in invoking the distinction (without prejudice, however, to whether such data might otherwise be explained). Some philosophers might want to include in set III.…

   blog/plato-stanford-edu/the-analytic-synthetic-distinction.md

8. 08 · blog0.747

   On the basis of Manders’ analysis he formulates three conditions necessary for a diagrammatic proof practice to be rigorous: a) it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object; b) the information thus displayed is not metrical; and c) it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. De Toffoli conceptualizes the diagrams of a diagrammatic proof method as a kind of notation in 2022 . She terms them mathematical diagrams and advances the following as a ‘Carnapian expl…

   blog/plato-stanford-edu/diagrams-and-diagrammatical-reasoning.md

9. 09 · blog0.746

   It has long been noted by commentators that mathematical proofs work with a particular case through universal instantiation ( ekthesis ) and then universalize to the general claim, and that not all propositions have the form: A is said of B , e.g., Elements 1 1, “To construct an equilateral triangle on a given line.” A more modern objection is that the formal theory of the syllogism as presented in Prior Analytics 1 1, 3-7 is woefully inadequate to express a theory involving conditionals and many-many relations, as is the case with all ancient mathematics. Nonetheless, Aristotle does think tha…

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

10. 10 · blog0.745

    The problem is grounded in the general idea that we project predicates onto reality (a reality that is itself “constructed” by those projections, according to the constructivist approach Goodman defended from the time of A Study of Qualities [1941], hence in The Structure of Appearance [1951] and, later, in Ways of Worldmaking [1978a]). Hume famously claimed that inductions are based on regularities found in experience, and concluded that the inductive predictions may very well turn out being false. In Fact, Fiction, and Forecast , Goodman points out how “regularities” are themselves in a sens…

    blog/plato-stanford-edu/goodman-s-aesthetics.md