seen from India
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woh woh formal sciences time
Logic & Computer Science
Though Computer Science can be referred as—well Computer science,it's best to refer to these guys as (The) Computer Sciences since they are a 3-in-1 system.In other words,organism.But they are not biologically alive aren't they?
I also notice that so far for the Formal Sciences,2/3 of them are monochrome colored lmao.Math does have some golden accents to them but they are mostly monochrome colored
I HAVE A SKIT
Biology: This snake is 3.14 meters long,impressive for it's genus!
Mathematics: Heh,I guess you can say...πthon.
Biology: What?
The Auxiliary Sciences of Formal Thought
The Penumbra of Logic.—Outside the boundary of logic proper, inhabiting the penumbra of logic, if you will, are theories of truth, the nature and role of definitions, and the structures of formal systems. Other modes of formal thought, such a mathematics and mereology, also employ or imply these penumbric ideas. Axiomatics, being the paradigmatic structure of a formal system, has been so thoroughly assimilated into mathematics it is often taken to be a part of it, but the axiomatic method—long called the geometric method because of its association with the exposition of geometry since Euclid—is equally applicable to any of the formal sciences, and indeed is employed in the natural sciences as well when rigor appears to demand it. But axiomatics didn’t appear in a work on logic until the Port-Royal Logic in the seventeenth century (La logique, ou l’art de penser, 1662). Nevertheless, these auxiliary sciences of formal thought have a closer relationship to logic than the other formal sciences, as various systems of logic down through history have included them or excluded them as the logic in question was conceived with a narrower or wider scope. Aristotle gives his theory of doctrine of definition in the Posterior Analytics, so if we take this book to be a part of logic (and it is traditionally part of the Organon), then definition is a part of logic. But the Posterior Analytics is widely recognized as a work in the theory of science (at least, science in the Platonic sense of demonstrative knowledge), so that the position of this work within logic itself is the question of the position of definition within logic writ large. Being shared, then, among the formal sciences, serving as the common language of the formal sciences, are these disciplines in the penumbra of logic actually the essence of formal thought? Rather than being in the gray area of formal thought, ought they to stand out as the most sharply defined exemplars of formal thought?
Brief Addendum on the Overview Effect in Formal Thought
Some time ago on my other blog in The Overview Effect in Formal Thought I discussed the intuitive significance of seeing things whole, which in the context of formal thought means the use of a concise notation that allows one to see in one glance the idea expressed by the symbolism.
On this I cited a quote from Russell and Whitehead’s Principia Mathematica:
“The terseness of the symbolism enables a whole proposition to be represented to the eyesight as one whole, or at most in two or three parts divided where the natural breaks, represented in the symbolism, occur. This is a humble property, but is in fact very important in connection with the advantages enumerated under the heading.” (Bertrand Russell and Alfred North Whitehead, Principia Mathematica, Volume I, second edition, Cambridge: Cambridge University Press, 1963, p. 2)
A discussion today on the FOM (foundations of mathematics) listserv made me aware that there is a temporal as well as a spatial component to the overview effect in formal thought. A notation must not only represent a proposition, “to the eyesight as one whole,” as Russell says, but it must also allow the mathematician to develop his ideas rapidly and efficiently without being so unwieldy that it slows down the process of thought, hampering the formal expression of an idea rather than facilitating the same.
One might reformulate the quote above in terms of temporality such that, the terseness of the symbolism enables the development of an exposition and the proof of a theorem so that it is represented to intuition as one whole in time. This is a very different approach to the intuition of time in formal thought than that which Brouwer employed as the basis of intuitionism.
I am reminded of a passage from Bertrand Russell’s Autobiography in which he described Alfred North Whitehead, his co-author on Principia Mathematica:
“His capacity for concentration on work was quite extraordinary. One hot summer's day, when I was staying with him at Grantchester, our friend Crompton Davies arrived and I took him into the garden to say how-do-you-do to his host. Whitehead was sitting writing mathematics. Davies and I stood in front of him at a distance of no more than a yard and watched him covering page after page with symbols. He never saw us, and after a time we went away with a feeling of awe.”
Whitehead, apparently, had a symbolic notation sufficient to his purposes and allowing him to cover page after page in symbols, recording his thought as he went. Many are the philosophers of mathematics who have pointed out that the level of formal rigor in Russell and Whitehead’s Principia Mathematica fell short of the standard that Frege had already employed some years previously in his Grundgesetze der Arithmetik. While this cannot be denied, there may be a point to a less-than-perfect rigor which nevertheless allows for rapidity and fluency of expression.
It seems that logic and mathematics require just enough formalization in order to be fruitful, but not too much. I once wrote, sufficient unto the day is the rigor threreof. Perhaps I should have written, sufficient unto the moment is the rigor thereof. Think of this as one aspect of the ethos of formal thought.
The second complex analysis lesson is up! Learn yourself some math if you dare!
The first complex analysis lesson is up! Download it if you dare.
