Summa minutiae

“A facility for quotation covers the absence of original thought.” —Lord Peter Wimsey

Posts tagged "math":

Conservation of meaning

So I'm considering going back and working through all the math that I zipped through in school but never deeply understood. Act 1: the start of George Chrystal's Algebra: An Elementary Text-Book, the 1904 Adam & Charles Black edition.

My comments and notes are in quotation blocks, like this.

Chrystal's Algebra, chapter 1, section 1.

The student is already familiar with the distinction between abstract and concrete arithmetic.

I don't have a syllogism handy, but if you run something past me I can probably figure out how to agree with it since that's how this whole textbook idea works.

The former is concerned with those laws of, and operations with, numbers that are independent of the things numbered; the latter is taken up with applications of the former to the numeration of various classes of things.

concrete arithmetic: regards the numeration of actual specific things; application of the laws of number and operations performed on numbers. Use of numbers to count things.

abstract arithmetic: the same laws and operations as they concern numbers themselves. Use of numbers in and of themselves.

It's been a long time since arithmetic and I don't recall that particular distinction, but OK.

In the spectrum of math from "lower" to "higher":

number -> concrete arithmetic -> abstract arithmetic

Confining ourselves for the present to abstract arithmetic, let us consider the following series of equalities :—

\[\begin{align} \frac{2623}{61} + \frac{1023}{3} &= \frac{2623 \times 3 + 1023 \times 61}{61 \times 3} \\ &= \frac{70272}{183}\\ &=384.\\ \end{align}\]

The first step is merely the assertion of the equivalence of two different sets of operations with the same numbers. The second and third steps, though doubtless based on certain simple laws from which also the first is a consequence, nevertheless require for their direct execution the application of certain rules, of a kind to which the name arithmetical is appropriated.

Step 1: we've thrown numbers and operators around in a particular way and then we claim (assert?) that the two piles of rearranged stuff have the same meaning.

Steps 2 and 3: whoa - mysterious rules of the operators were applied to transmogrify these piles of stuff into entirely different things, and then we assert that they all have exactly the same meaning. The visible forms of these things were changed radically, but their meanings were left perfectly untouched.

So we're given a thing, and we want to change it into something easier to understand or easier to work with, but there is a limited menu of possible changes.

The artistry lies in knowing the menu, selecting the right items from it, and using them in just the right way to change the appearance of the thing without changing its meaning. Maybe you could see it as a law of "conservation of meaning" that must be obeyed at every step. If applying a menu item would result in a different meaning, then you can't use it. Unless you're fluent enough to then select another item from the menu that will save the meaning.

We have thus shadowed forth two great branches of the higher mathematics:—- one, algebra, strictly so called, that is, the theory of operation with numbers, or, more generally speaking, with quantities; the other, the higher arithmetic, or theory of numbers. These two sciences are identical as to their fundamental laws, but differ widely in their derived processes. As is usual in elementary text-books, the elements of both will be treated in this work.

Note the dog's bollocks that were commonly used in British typography way back when. For more information:– wikipedia, stackexchange, Nick Martens (who cites hellbox.org).

math test

When \(a \ne 0\), there are two solutions to \(ax^2 + bx + c = 0\) and they are \[x = {-b \pm \sqrt{b^2-4ac} \over 2a}.\]

\(\sin(z)\) has a series expansion \(\sum _{k=0}^{\infty } \frac{(-1)^{k-1} z^{2 k-1}}{(2 k-1)!}\).