A comment in an online forum thread on my post on Laudato Si’ notes the following about mathematical science:

the application of mathematics to nature is not a particularly 17th century thing, but it goes to the beginnings of civilization, to the point that we could even say that mathematics is natural to the mind and making maths is of the same cultural necessity as making music (an activity that, although strictly not necessary, is present in every civilization).

This is obviously true, and the brief account of the new-science in my post wasn’t meant do deny it, but I can see how the post could be read that way.

What *is* particular to the sort of mathematical application found in the new science of the 17th century is the homogenizing character of the mathematical symbolization used, and the elevation of this application to a *mathesis universalis, *a universal method which becomes, *the *dominant approach to reality. I have posted a long explanation of what that means (from a dissertation draft) to academia.edu. It’s the fullest account of the scientific revolution that I have written so-far, but it is still only a partial account. I am planning to expand it into a full-blown treatise at some point.

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I admit I am having a difficult time digesting the critiques of algebra and Descartes. Is it that algebra is vaguely immoral and should not be taught to impressionable children in schools? Or is it that the notation and method of discourse should be redesigned to give a clearer referent to ‘unknown’ quantities in algebraic expressions (or whatever equivalent)? What are the implications of taking seriously the Greek distinction between magnitude and number? Are there any, or do we just rename ‘natural numbers’ into ‘numbers’, ‘rational numbers’ into ‘ratios’, ‘real numbers’ into ‘magnitudes’, and go on exactly as we did before?

To be honest, I think the criticism that unknowns in algebra are an unwarranted layer of abstraction is off base. You claim that mathematicians do the following with the variable ‘a’:

“One might say that it intends a class of numbers, but it is not treated as class; in the equation it is treated as though it is a determinate number.”

One of the first things I learned dealing with algebra is that this is surely the way to become no kind of mathematician at all. Algebraic expressions ought to be treated as if the variables are quantified in a definite fashion. In practice quantifiers are omitted left and right (arguably a bad habit but, as Descartes says, it serves to ‘shorten the work’ considerably), but they *cannot* be omitted from one’s understanding. (How omitted variables are quantified may be ‘obvious’ in some contexts as a matter of convention and this feature of mathematical discourse might be considered problematic — but it is no more problematic than any other human communication which depends on cultural context.) Otherwise the expression is strictly meaningless and cannot be manipulated at all — you have no clue what manipulations are valid. You cannot “multiply both sides” of an inequality by some symbol unless you know that the symbol definitely intends a positive number, or a class of positive numbers.

The way that algebra is taught in most schools fails to get this point across, so that many students form the impression that algebra is a question of ‘manipulating abstract equations according to some arbitrarily defined rules’. To be honest in that sense the modern curriculum is kind of immoral. But that and the intellectual validity of algebra as a way of understanding relations between magnitudes are two different things.

If you want to produce a more thorough critique of symbolic calculation you would also do well to comment on more central things to mathematics. The attached paper immediately raises the question of how, absent symbolic calculation, mathematicians are to talk about *functions* — the questions of what a function is, why they are important to mathematics (or perhaps not?), whether functions are a real entity that the Greeks happen to have neglected or a fever dream of post-Cartesian mathematicians, is entirely absent from the discussion. Set theory is also an important topic (and functions are more directly grounded in set theory than as a way of describing curves on a Cartesian plane).

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http://math.stackexchange.com/questions/348198/best-fake-proofs-a-m-se-april-fools-day-collection

The fact that mathematical symbols have a definite referent is most quickly demonstrated by inspecting ‘joke proofs’ of clearly false propositions which are obtained by equivocating on what entity some symbol refers to. The ‘no dogs’ proof is a good example of what such a proof reads like when translated to a verbal syllogism.

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I don’t think that algebra is directly immoral, but rather that it serves as the foundation of a kind of science, dominant in our day, that makes people blind to the intrinsic goodness and teleology of things, and that therefore leads down the line to their inability to understand natural law.

I recommend Harvey Flaumenhaft’s paper on these questions in the SJR: https://www.sjc.edu/files/7113/9657/8086/sjc_review_vol48_no1.pdf Here’s a snip:

«Mathematical modernity gets under way with Descartes’ Geometry. By homogenizing what is studied, and by making the central activity the manipulative working of the mind, rather than its visualizing of form and its insight into what informs the act of vision, Descartes transformed mathematics into a tool with which physics can master nature… For those who study Euclid and Apollonius in a world transformed by Descartes, many questions arise: What is the relation between the demonstration of theorems and the solving of problems? What separates the notions of how-much and how-many? Why try to overcome that separation by the notion of quantity as represented by a number-line? What is the difference between a mathematics of proportions, which arises to provide images for viewing being, and a mathematics of equations, which arises to provide tools for mastering nature? How does mathematics get transformed into what can be taken as a system of signs that refer to signs—as a symbolism which is meaningless until applied, when it becomes a source of immense power? What is mathematics, and why study it? What is learning, and what promotes it?»

You write, “You cannot “multiply both sides” of an inequality by some symbol unless you know that the symbol definitely intends a positive number, *or a class of positive numbers.*” That’s the point; a class of numbers is not a definite number.

I think you are right that the question of functions is important here.

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Thank you for the reference to the SJR articles. I would point out again that, strictly speaking, an algebraic equation is meaningless in and of itself. Thus, “y = |x| + 2” could be used in any of the following contexts:

“Given a definite number y (say, y = 3), consider the set of those numbers x which satisfy the relation y = |x| + 2.”

“The set of those pairs of numbers (x, y) which satisfy the relation y = |x| + 2.”

“The function y = |x| + 2.” (The function which maps any given number, which we denote x, into the absolute value of said number x plus two. This result we denote y.)

Thus until we have made exactly clear which classes x and y are drawn from, we have no idea what the equation relating them is talking about.

It occurs to me that functions would be particularly important because they are bound up closely with a genuine revolution in consciousness that might be problematic. As one of the SJR contributors (Michael N. Fried) claims, the Greeks were averse to making motion into a subject-matter of their demonstrations (albeit motion related language might crop up in a sort of ‘origin-myth’ detailing the construction of a figure that becomes the subject of a theorem). In this respect modern mathematics has not really overthrown the Greek view of things, so much as figured out a way to expand its scope. The revolution in consciousness involved in mathematical description of motion was that, in order to obtain a static *object* that can be analyzed, time is reduced to yet another dimension of space, and therefore made the subject of geometry. That this reveals some kind of correspondence between the geometrical notion of a tangent line — and the physical notion of velocity — was a deeply unexpected notion. This is kind of important because it may even be a factor influencing how we perceive relatively more ancient philosophical accounts of a supra-temporal existence. (Did ancient theologians really think of God as being outside of time in a way that had anything to do with how mathematicians since Leibniz have thought of the motion of an object being described by a static curve?) Nor is this revolution, strictly speaking, dependent on algebra — we could describe the back-and-forth oscillation of a spring as a geometrical curve (with one space and one time dimension), and them make its properties the subject of a geometrical demonstration in a style entirely consistent with the conventions of Greek mathematics.

In any case, I’ll look forward to a fuller account of your views, even if I expect to disagree with them.

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