Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.
–Bertrand Russell, Recent Work on the Principles of Mathematics, published in International Monthly, Vol. 4 (1901), courtesy of Wikiquote
We left off last time with hints of fascinating implications of the idea of God making the universe by His emanations. To refresh, we noted that “creation” implies making something ex nihilo–out of nothingness. The thing so created is existentially separate from God, although, according to Thomist thought, at least, it requires His ongoing action to continue in existence. By contrast, “emanation” implies a “flowing into”, whereby a “part” of God “flows into” what He makes, bringing it into existence. In a sense, the things emanated are not existentially or ontologically separate from God. Let’s look at this latter mode of making a universe in more detail.
First, it’s necessary to point out that no amount of emanation–no amount of “flowing out”–ever exhausts God’s essence. A reservoir has a limited amount of water to flow out into irrigation channels, to households, and so on. Let out too much and it will be left dry. Likewise, if I keep pinching smaller lumps of clay off of a larger lump, sooner or later there will be no large lump left. It doesn’t work that way with God, though–there’s no limit to the number of beings or entities (the technical theological term is “creature”–we use it to mean animals, but literally, “creature” means anything, animal, vegetable, or mineral, that has been created. It is in this sense that I’ll use the term here) which He can emanate. This is simply because God is infinite.
… es ist wahr, ein Mathematiker, der nicht etwas Poet ist, wird nimmer ein vollkommener Mathematiker sein.
… it is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician.
–Karl Weierstrass, letter to Sofia Kovalevskaya, August 27, 1883, as shared by Gösta Mittag-Leffler at the 2nd International Congress for Mathematicians in Paris. Compte rendu du deuxième Congrès international des mathematiciens tenu à Paris du 6 au 12 août 1900, Gauthier-Villars (Paris), 1902, page 149. Courtesy of Wikiquote.
Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of Elements.
–As quoted in “A Paper of Omar Khayyam” by A.R. Amir-Moez in Scripta Mathematica 26 (1963). This quotation has often been abridged in various ways, usually ending with “Algebras are geometric facts which are proved”, thus altering the context significantly. Courtesy of Wikiquote.
I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him.
Treatise on Demonstration of Problems of Algebra (1070). Courtesy of Wikiquote.
By the help of God and with His precious assistance, I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to “things” which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of Algebra as stated above. The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns.
–Treatise on Demonstration of Problems of Algebra (1070). Courtesy of Wikiquote.
The first of a series of three Quotes of the Week from our Patron, from works other than the Rubáiyát.
This is an essay originally written about seven years ago. The subject matter is just as pertinent, if not more so.
Over the last month I have been working every other day as a substitute teacher. I am currently working towards secondary school certification so I can teach full-time in the public schools at the high school level. Of course, I’ve taught for seventeen years, but I still need that piece of paper for the public schools…but don’t get me started on that…. Anyway, most recently I taught for six years at a proprietary (i.e. for-profit) two-year college. It was during this time that I discovered that proprietary colleges are the tool of Satan. Don’t get me started on that, either. Anuway, as you may guess, doing the proprietary college thing didn’t work out, and it’s time for a change. Thus, partly for continued income (always a good thing, especially if you’ve got a mortgage, a toddler, and four cats!) and partly to get my foot in the door, I’ve been subbing at the high school and middle school in the town where I live. Next semester, when I will be taking all night classes at college, I hope to substitute every day, rather than every other day, as now. Meanwhile, even subbing every other day has provided some interesting insights.
My field is mathematics–let me point that out right away. Also, let me point out a few other things. First, I’m a mediocre mathematician–I’m not even fit to unlace John Nash’s shoes, mathematically speaking. I’m a damn good math teacher though. Though I actually am an arrogant bastard, that’s not boastful blather, but a reasonable hypothesis based on seventeen years of consistently good performance evaluations–except for one from the aforementioned Satanic college, which according to a friend in the know there, was political; but don’t get me started on that!–and testimonials by former students. I know my weaknesses, though, and as I said, I am a mediocre mathematician. I graduated from a state university not known as a training ground for great mathematicians, and before that I graduated from a high school of only about 800 (graduating class of 160). I was raised in a very small town of a population roughly equal to the student body of my high school (which was, obviously, in a different town). The point is, mathematically, both in aptitude and training, I am nothing special, to say the least. Read the rest of this entry
Mathematics, rightly viewed, possesses not only truth, but supreme beauty —a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.