Workshop 1Foundations of Deductive Reasoning

1. Are there limits to what humans can know?2. Are there limits to computer knowledge?

Overview

Euclid’s Elements

Kepler’s laws of planetary rotation

Newton’s laws of motion and gravity

Leibniz’s computer

Hilbert's challenge

Hilbert’s axioms for geometry

Principia Mathematica - Bertrand Russell & Alfred North Whitehead

The Antikythera Mechanism – A Greek Computer?

Freeth, Tony; Jones, Alexander (2012). "The Cosmos in the Antikythera Mechanism". Institute for the Study of the Ancient World. Retrieved 19 May 2014.

NOVA | Ancient Computer - PBS

www.pbs.org/wgbh/nova/ancient/ancient-computer.html

Euclid’s elements

Axioms1. Things which are equal to the same thing are also equal to one another.2. If equals are added to equals, the whole are equal.3. If equals be subtracted from equals, the remainders are equal.4. Things which coincide with one another are equal to one another.5. The whole is greater than the part.

Postulates 1. To draw a line from any point to any point.2. To produce a finite straight line continuously in a straight line.3. To describe a circle with any center and distance.4. That all right angles are equal to one another.5. That, if a straight line falling on two straight lines make the interior

angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side of which are the angles less than the two right angles.

Proving a Theorem (a construction)Proposition 1: To construct an equilateral triangle on a given finite straight line.

Let AB be the given finite straight line.

• To construct an equilateral triangle on the straight line AB. • Draw the circle BCD with center A and radius AB. Draw the

circle ACE with center B and radius BA. Join the straight lines CA and CB from the point C at which the circles cut one another to the points A and B. Postulates 3,1

• Now, since the point A is the center of the circle CDB, therefore AC equals AB. Again, since the point B is the center of the circle CAE, therefore BC equals BA. Definition 15

• But AC was proved equal to AB, therefore each of the straight lines AC and BC equals AB.

• And things which equal the same thing also equal one another, therefore AC also equals BC. Common Notion 1

• Therefore the three straight lines AC, AB, and BC equal one another.

• Therefore the triangle ABC is equilateral, and it has been constructed on the given finite straight line AB. Definition 20

• Q.E.D.

From Euclid's ElementsDr. David E JoyceClark University

Kepler's laws of planetary motion

1. The orbit of a planet is an ellipse with the Sun at one of the two foci. (1609)

2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. (1609)

3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. (1619)

• The sun’s gravity accelerates the planet as it moves towards the sun, slows it as it moves away.

• It sweeps out a big piece of arc on the right and a small one on the left

• The areas swept out are the same.

Newton’s Laws of Motion

Newton's law of universal gravitation

Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.

Deriving Kepler’s Laws

Kepler’s laws of planetary rotation are drivable from Newton’s laws of motion and gravity, a triumph of the deductive approach.

1684 De motu corporum in gyrum ("On the motion of bodies in an orbit"). Manuscript by Isaac Newton sent to astronomer Edmond Halley.

Click above link to see a derivation as in Euclid’s elements.

After they had been sometime together, the Dr [Halley] asked him what he thought the curve would be that would be described by the planets …. Sir Isaac replied immediately that it would be an Ellipsis, the Dr struck with joy & amazement asked him how he knew it, Why saith he I have calculated it. (According to Abraham de Moivre)

Pascal’s Pascaline

Blaise Pascal invented the Pascaline, a mechanical calculator to aid his father tabulate tax collections

It could add and subtract A Pascaline from 1652 (© Musée des Arts et Métiers, Paris)

Leibniz’s Automata

Leibniz saw a hierarchy of elementary divine automata (monads) as representing the universe1.

Thus the organic body of each living being is a kind of divine machine or natural automaton, which infinitely surpasses all artificial automata. For a machine made by the skill of man is not a machine in each of its parts. For instance, the tooth of a brass wheel has parts or fragments which for us are not artificial products, and which do not have the special characteristics of the machine, for they give no indication of the use for which the wheel was intended. But the machines of nature, namely, living bodies, are still machines in their smallest parts ad infinitum. It is this that constitutes the difference between nature and art, that is to say, between the divine art and ours2

1. Klaus Mainzer, Leon Chua, The Universe as Automaton: From Simplicity and Symmetry to Complexity, Springer 2012

2. Leibniz, GW. , Monadology, 1714

Leibniz’s Calculus Ratiocinator

Reasoning is nothing but the joining and substitution of characters, whether these characters be words, symbols or pictures

Atomic (elementary) concepts can be represented by prime numbers

The truth value of statements can be calculated arithmetically

The calculation could be automated using a computer

The Stepped Reckoner of Gottfried Leibniz

Gottfried Wilhem Leibniz – video cacheLeibniz sketch of a stepped-drum computer

Leibniz Calculating Machine

Replica of the Leibniz calculating machine manufactured by HNF, 1995, according to the reconstruction by Prof. Dr. N. J. Lehmann

Paderborn Computer Museum Exhibits and background of computer pioneers

Ignoramus et Ignorabimus

Proposed Seven World Riddles 1880, the three below he considered unsolvable

1. the ultimate nature of matter and force2. the origin of motion3. the origin of simple sensations

"ignoramus et ignorabimus" "we do not know and will not know."

Emil du Bois-Reymond (1818 –1896) German physician and physiologist

David Hilbert

We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus.

For us there is no ignorabimus, and in my opinion none whatever in natural science.

Our slogan shall be:

David Hilbert 1862 - 1943

Wir müssen wissen wir werden wissen!

We must know we will know!

Hilbert’s Axiomatization of Geometry

Attempt to fix Euclidean geometry 1899

Sample axioms:

For every two points there exists no more than one line that contains them both

If A and C are two points of a line, then there exists at least one point B lying between A and C.

Hilbert's Program (1920s)- called for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is

consistent. John von Neumann contributed

Kurt Gödel’s incompleteness theorems later showed Hilbert's Program cannot be carried out.

Principia Mathematica - Bertrand Russell & Alfred North Whitehead 1910 – 1913

Bertrand Russell - Face to Face Interview (BBC, 1959) Delightful interview though not much on math or logic

• An attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven.

• 1931 Kurt Gödel demonstrated this could not be achieved.

Kurt Gödel and the Mathematical Paradox | "This Statement is unprovable (2 minutes) cache

In any non-trivial axiomatic system, there are true theorems that cannot be proven.

Discussion

Why are children taught geometry?

Leibniz believed living bodies are still machines in their smallest parts ad infinitum. Was he right given what we now know about biology?

Why was Hilbert so confident when he said “We will know”?

Was Hilbert right?

Can the stock market be predicted if we had enough data?

Friedrich Engels described Marx’s theories as “scientific socialism” as opposed to the utopian socialism which preceded it. Is the idea that social and political changes are driven by economics comparable to Newton’s deriving Kepler’s laws from his own more fundamental ones?