Below are some quotations that are related to material discussed in class or the course in general.

May they stir your curiosity!

Note: The quotations below have been typed in by myself or taken from various places on the internet. They should serve only to give an idea of the author's views and a reference for where to find it expressed. Thus, please do not quote these lines, but check first with a reliable source!

- Socrates on knowledge:
So I examined this man-there's no need for me to mention his name, let's just say he was a politician-and the result of my examination and of my conversations with him, was this. I decided that although the man seemed to many people, and above all to himself, to be wise, in reality he was not wise. I tried to demonstrate to him that he thought he was wise, but actually was not, and as a result I made an enemy of him, and of many of those present. To myself, as I left him, I reflected, "Here is one man less wise than I. In all probability neither of us knows anything worth knowing; but he thinks he knows when he doesn't, whereas I, given that I don't in fact know, am at least aware I don't know. Apparently, therefore, I am wiser than him in just this one small detail, that when I don't know something, I don't think I know it either." From him I went to another man, one of those who seemed wiser than the first. I came to exactly the same conclusion, and made an enemy of him and of many others besides.

(Plato,

*Apology*, 21c) - Aristotle on truth:
To say of what is that it is not, or of what is not that it is, is is false, while to say of what is that it is, or of what is not that it is not, is true.

(

*Metaphysics*, 1011b32 ff.)

**Mathematical reasoning** - On Galileo's use of mathematical reasoning:
Galileo's chief contributions to the new natural philosophy are his insertion of mathematical reasoning and observation into arguments about the nature of the world.

(David Harley, http://www.nd.edu/~dharley/HistIdeas/Galileo.html)

- On Descartes' use of the mathematical method:
On the night of 10 November, 1619 Descartes experienced an intellectual enlightenment. As a result, he conceived the idea of a complete reform of philosophy, based on the methods of mathematics.This event was the culmination of reflection inspired by Isaac Beeckman, whose chance meeting with Descartes the previous year marked the beginning of a long friendship.

[...] Following his enlightenment of 1619, Descartes attempted to apply to physics the method of mathematics, which was free from the obscurity of principle and sterility of deduction that he discerned in scholastic philosophy.

(Aiton,

*The Vortex Theory of Planetary Motions*, p.30)

**Inductive reasoning** - David Hume on inductive reasoning:
All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.

Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind.

(David Hume (1711-76).

*An Enquiry Concerning Human Understanding*(1748). Chapter IV:*Sceptical Doubts concerning the Operations of the Understanding*, Part I.)

**On Euclid** - Thomas Hobbes on Euclid:
He was 40 years old before he looked in on Geometry; which happened accidentally. Being in a Gentleman's Library, Euclid's Elements lay open, and 'twas the

**47 El. libri I**. He read the Proposition. By God, sayd he (he would now and then swear an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et**sic deincep**s that at last he was demonstratively convinced of that trueth. This made him in love with Geometry.(From the life of Thomas Hobbes in John Aubrey's

*Brief lives*, about 1694.

Quote copied from www.sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagoras.html.) - Betrand Russell (1872-1970) on Euclid:
At the age of eleven, I began Euclid, with my brother as tutor. This was one of the great events of my life, as dazzling as first love.

(Quoted from Dunham,

*Journey through genius*, p.~31.) - Albert Einstein on Euclid:
At the age of 12 I experienced a second wonder of a totally different nature: in a little book dealing with Euclidean plane geometry, which came into my hands at the beginning of a schoolyear. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which --- though by no means evident --- could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me. For example I remember that an uncle told me the Pythagorean theorem before the holy geometry booklet had come into my hands. After much effort I succeeded in ``proving'' this theorem on the basis of the similarity of triangles ... for anyone who experiences [these feelings] for the first time, it is marvellous enough that man is capable at all to reach such a degree of certainty and purity in pure thinking as the Greeks showed us for the first time to be possible in geometry.

(From pp. 9-11 in the opening autobiographical sketch of

*Albert Einstein: Philosopher-Scientist*, edited by Paul Arthur Schilpp, published in 1951.

Quote copied from www.sunsite.ubc.ca/DigitalMathArchive/Euclid/java/html/pythagoras.html.) - Edna St Vincent Millay (1892-1915):
*Euclid alone has looked on beauty bare*Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing, intricately drawn nowhere

In shapes of shifting lineage; let geese

Gabble and hiss, but heroes seek release

From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,

When first the shaft into his vision shone

Of light anatomized! Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.(Brought to my attention by Eunice Liew.)

**Contradictions** - Russell's letter to Frege (6/16/1902), announcing that he found a
contradiction in Frege's work:
I have known your

*Grundgesetze der Arithmetik*for a year and a half, but only now have I been able to find the time for the thorough study I intend to devote to your writings. I find myself in full accord with you on all main points, expecially in your rejection of any psychological element in logic an in the value you attach to a*Begriffsschrift*for the foundations of mathematics and of formal logic, which, incidentally, can hardly be distinguished. On many questions of detail, I find discussions, distinctions and definitions in your writings for which one looks in vain in other logicians. On functions in particular (S9 of your*Begriffsschrift*) I have been led independently to the same views even in detail. I have encountered a difficulty on on one point. You assert (p.17) that a function could also constitute the indefinite element. This is what I used to believe, but this view now seems to me dubious because of the following contradiction: Let*w*be the predicate of being a predicate which cannot be predicated of itself. Can*w*be predicated of itself? From either answer follows its contradictory. We must therefore conclude that*w*is not a predicate. Likewise, there is no class (as a whole) of these classes which, as wholes, are not members of themselves. From this I conclude that under certain circumstances a definable set does not from a whole.(Source: Michael Beaney (ed.),

*The Frege Reader*, Blackwell Publishers, Oxford, 1997; p.253). - Frege's reply to Russell (6/22/1902):
Dear Colleague,

... Your discovery of the contradiction has surprised me beyond words and, I should almost like to say, left me thunderstruck, because it has rocked the ground on which I meant to build arithmetic.

... It is all the more serious as the collapse of my law V seems to undermine not only the foundations of my arithmetic but the only possible foundation of arithmetic as such.

... Your discovery is at any rate a very remarkable one, and it may perhaps lead to a great advance in logic, undesirable as it may seem at first sight.(Source: Michael Beaney (ed.),

*The Frege Reader*, Blackwell Publishers, Oxford, 1997; p.254).

- Dirk's teaching motto:
Good college teaching is the kind that promises to make the teacher finally superfluous, the kind that will lead students to want to continue work in the given subject and to be able to have the necessary intellectual equipment to continue work at a more advanced level.

(Wayne Booth,

*What Little I Think I Know about Teaching*)

© Dirk Schlimm, Last modified: 2/15/02