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Leonard Peikoff raised a comparison in his assessment of a plague, being not fully exact in his short essay "The Analytic-Synthetic Dichotomy", (TA-SD, for short.)

A plague attacks man's body, not his conceptual faculty. And it is not launched by the profession paid to protect men from it.

Analytic and synthetic have quite a different connotation in geometry. Having spent four decades cutting sections, projecting views, the latest segment of this adventure has segued into dimensional analysis that takes each feature and considers the range of variation permitted it in size and location, often in relationship to a corresponding size and relationship to ensure fit, finish and functionality.

Philosophy has lauded mathematics as the epitome of certainty in the past. Geometry is a branch of mathematics that is further subdivided into an analytic and synthetic branch that has operated synergistically in the application described in the previous paragraph.

From the 1913 Webster, 


(a.) Not original or primary; received from another.

(a.) Not new; already or previously or used by another; as, a secondhand book, garment.

While there is little new in geometry in the earlier described approach, it helps to grasp it for one's self in order to bridge from the synthetic to the analytic without introducing a logical dichotomy. In a world where the use of computerized automation is expanding, the volume of firsthand knowledge needed brings people of various disciplines together in order to provide as seamless an integration as can be mustered in an age "where each man must be his own intellectual protector." (TA-SD)

The synthetic side of geometry combined with the analytic side on the drafting board made a logical leap (pardon the pun) to the computerized "drafting board" during my career. An interest in philosophy introduced Peikoff's juxtaposition as well. 

Geometric Tolerancing and Dimensioning (GD&T) is a more recent approach to providing a standardized approach where communication gaps have cropped up. GD&T has been described as a design philosophy and has been used to aide in clarification and occasionally contribute to confusion when it comes to interpretation of a drawing's intent. It is interesting to note here that the standard bridges yet another issue, the issue Ayn Rand addressed in the book TA-SD is included in its second edition, the issue of concepts. 

Identifying math as the science of measurement, she sought to unravel the role it played concept formation. Measurement omission, or measurement inclusion, identified that characteristics varied within a range, but in the context of concept formation, were implicitly omitted. On a technical drawing, it is when a range is not specified explicitly that ambiguity is introduced. To reduce costly errors that can arise via unclear communication, organizations and committees have been formed, at a cost to address the issue of concepts in this particular application, although it is not explicitly identified as such. It is also done to provide a level of uniformity and approach across many languages, contributing to the value added to the undertakings.

In a concluding pun, the question of who will protect us from our protectors is somewhat apropos. 


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Thanks, Greg.

“Let no one unversed in geometry enter here” is said to have been inscribed over the entryway of Plato’s Academy. Excepting the value-philosophy areas, I think that Euclidean geometry should be a requirement, along with elementary logic, for any beginning philosophy student. I mean anyone beginning to study epistemology and metaphysics. Otherwise, one is missing necessary acquaintance with subject matter and tools of theoretical philosophy.

The synthetic and the analytic of the synthetic-analytic distinction in philosophy, from Leibniz, Hume, Kant, logical positivists, and linguistic analysts, are not the synthetic (e.g. Euclid) and the analytic (e.g. Descartes) of geometry.

On the synthetic-analytic distinction in philosophy, the notions and distinction argued against in Peikoff 1967, there are the recent works Truth in Virtue of Meaning (2008) by Gillian Russell and Analyticity (2010) by Cory Juhl and Eric Loomis. This distinction has continued to be a controversy among philosophers, though most weigh in against soundness of the distinction.

Books on my shelf pertaining to the quite different distinction between synthetic and analytic geometry (all these books still awaiting my full assimilation) are History of Analytic Geometry (1956) by Carl Boyer, The Development of the Foundations of Mathematical Analysis from Euler to Riemann (1970) by Ivor Gratton-Guiness, and Analysis and Synthesis in Mathematics (1997) edited by Michael Otte and Marco Panza.

Greg mentioned that the two distinctions of synthetic-analytic—in philosophy v. in geometry—are not equivalent distinctions. Though the concept of the analytic in geometry is not the same as the analytic in epistemology and although the synthetic in geometry is not the same as the synthetic in epistemology, the distinction from geometry can be of great import for contemporary metaphysics and epistemology. At least this looks so in my present large and main project.*

Edited by Boydstun
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6 hours ago, Boydstun said:

Excepting the value-philosophy areas, I think that Euclidean geometry should be a requirement, along with elementary logic, for any beginning philosophy student.

Why specifically Euclidean?


I'm not opposed to the idea that budding philosophers should have a good handle on the physical world they are to describe, per se; I'm just not sure why Euclidean geometry, specifically, is what you would connect to that.  But I know very little about mathematical geometry, myself, which is part of what makes me so curious.

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Harrison, I mention Euclidean because it is learnable in high school. It is a power of one's mind for knowledge quite different from the power of learning chemistry (including lab). By learning Euclid, I mean becoming able to prove things by the elementary assumptions or anyway understanding the proofs that are given in Euclid. The textbook at my high school was not Euclid himself.* It was the SMSG book, which worked really well.

You know, what would be particularly good to know are the proofs that (i) the sum of the angles of any triangle equal two right angles, (ii) the Pythagorean Theorem, and (iii) the diagonal of a unit square does not stand in any ratio to its sides. These are especially prominent for tries at epistemology from Plato to Kant to the present.

Edited by Boydstun
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Usually "Euclidean geometry" refers to geometry consistent with Euclid's asumptions.  Non-Euclidean geometry refers to geometry inconsistent with Euclid's assumptions, especially geometry in which the parallel postulate is altered.

Euclidean geometry is relatable to ordinary experience in a way that non-Euclidean geometry is not.

Most college-bound people study Euclidean geometry in secondary school, or at least it was that way when I was in secondary school.

Non-Euclidean geometry is a more esoteric subject which is typically studied by math majors in college.

Differential geometry is a more advanced subject.  Its prerequisites include multivariable calculus and linear algebra.  Even though I am a math major, I never studied it.  It is a prerequisite to understanding Einstein's general theory of relativity, and is therefore of interest to physicists as well as mathematicians.  It is too advanced and technical for the kind of general education requirement Boydstun is talking about.

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