by: John D. Barrow, read in 2004
vii Definition of 'mathematician': The person you don't want to meet at a party.
3 Definition of 'science': "The replacement of our observations of the world by abbreviations which retain some or all of the information instantiated in the world."
19 Mathematics is a religion by definition.
70 Language originated 40,000 years ago
71 Chart of population groups, languages, and genetic distance
115 Hilbert: Consistency implies incompleteness??
117 Rudy Rucker's dream of Kurt Goedel's death
119 Goedel's Theorem
119 Goedel's doctoral thesis: The Completeness of Predicate Calculus
120 Writing "The Principia" nauseated Bertrand Russell
121 Transfinite Induction
123 Goedel as a profound telepathic Platonist
123 Goedel's proof of the existence of God - Leibniz's positive and negative properties
124 Goedel believed in a separate cosmic mind
124 Goedel's theological world view: "We are of course far from being able to confirm scientifically the theological world picture ... What I call the theological worldview is the idea, that the world and everything in it has meaning and reason, and in particular a good and indubitable meaning. It follows immediately that our worldly existence, since it has in itself at most a very dubious meaning, can only be the means to the end of another existence. The idea that everything in the world has a meaning is an exact analogue of the principle that everything has a cause, on which rests all of science."
124 James Sylvester - Mathematics compared with consciousness
125 Skolem - Categorical systems, models and isomorphism
126 "Skolem...established that no finite set of axioms can characterize the natural numbers uniquely; there will always exist some structure which is not isomorphic to the natural numbers but which satisfies the same finite set of axioms."
126 "If, instead of restricting our mathematical theories to those which are generated from a finite list of specific axioms, we allow there to exist axiom schemes, which are equivalent to infinite lists of specific axioms, then the system of natural numbers can be characterized uniquely."
127 A case for emergence? I.e. generating new information beyond what you have or know:?
127 Leibniz: Music is unaware counting
127 Leibniz dreamed of resolving all human disputes with logic and rigor.
128 Leibniz: "Let us calculate, Sir."
128 Leibniz's 2 and 5 year plan for fulfilling his dream.
129 Giuseppe Peano
129 "Until very recently [Goedel's Incompleteness Theorem] loomed over the subject of mathematics in an ambiguous fashion, casting a shadow over the whole enterprise, but never emerging to make the slightest difference to any truly practical application of mathematics."
130 Jean Dieudonne: Logic and set theory do not form the "foundations" of mathematics.
130 The Bourbaki project: "[T]he last hopes of the formalists; axiomatics, rigour, and soulless elegance prevail; diagrams, examples, and the particular are all eschewed in favour of the abstract and the general."
131 Laurent Schwartz on specialists and generalists
132 Bourbaki: Mathematics is 'a human creation and not a divine revelation'.
132 Bourbaki's comparison of growth of mathematics to growth of a city. How about extending that to growth of anything?: organisms, ecosystems, universes,...?
133 Bourbaki responsible for "New Math"
134 Einstein: "How can it be that mathematics, being after all a product of human thought independent of existence, is so admirably adapted to the objects of reality?"
134 Bourbaki on the relationship between mathematics and reality.
135 Definition of information: "We can associate a quantity of information with the axioms and rules of reasoning that define a particular axiomatic system. We define its information content to be the size of the shortest computer program that searches through all possible chains of deduction and proves all possible theorems."
136 Chaitin, randomness, diophantine equations, and information.
137 An undecidable problem involving "large" and "small" sets.
138 Randomness and complexity.
139 I think a problem with Chaitin's approach is that the "shortest program" is dependent, in part, on the conscious programmer's ingenuity.
141 Mathematics and the mystery of harmony between mind and the physical world.
142 Adding new axioms changes everything in the original formal system.
142 The intellectual deception of formal mathematics.
143 Doubts about the "certainty" of formal mathematical results.
143 Mathematics is either inexact or it rests on a super-human foundation.
144 A case for the universe being the consequence of some formal system.
144 Formal mathematics and Logical Positivism - Goedel, Popper
146 I disagree with his assumption that consciousness can emerge from a string of symbols.
146 Barrow: "Formalism is found lacking in two crucial respects. It fails to account adequately for the relationship between mathematical symbols and the minds of mathematicians and it fails to explain the utility of mathematics in describing the workings of the physical world." I think I can explain both of those things.
147 Barrow: Kant: "The very act of understanding adds something to the nature of the reality and so creates a gap between the real world and the perceived world."
148 Imagination and simulation are crucial aspects or capabilities of consciousness.
153 Shannon's Theorem.
155 Models and modeling
160 Symmetries and conservation laws
160 Gauge Theories
161 Superstring theory rules out infinity.
162 Symmetry breaking
162 Two flavors of the argument from design
163 What guarantees the existence of incompressible sequences??
169 Julian Jaynes
181 Ramanujan and the number 1729.
185 Luitzen Brouwer - intuitionism and three-valued logic
185 Brouwer denied 'infinity'.
187 Three-valued logic and quantum uncertainty
187 Penrose and Hawking's singularity theorems
188 Bertrand Russell: "The point of philosophy is to start with something so simple as to seem not worth stating, and to end with something so paradoxical that no one will believe it."
189 Hilbert's dismissal of the methods of Kronecker, Poincare, Brouwer, Russell, and Whitehead.
191 It seems that my views are most in line with Brouwer's.
195 Hermann Weyl defected to Brouwer's point of view.
196 Brouwer's intro to his philosophy is a lot like my Primordial Consciousness cosmology.
197 Brouwer: "Mathematics is nothing more, and nothing less, than the exact part of our thinking."
197 Bohr, split consciousness, quantum reality, and three-value logic
198 Kant's two categories of perception: spatial and temporal. This makes sense if you interpret the mind to be that of Primordial Consciousness rather than human. Euclidean geometry may still be correct if you consider extra dimensions in which to embed non-Euclidean manifolds.
199 Cantor vs. Kronecker; Hilbert vs. Brouwer
200 I think Kronecker was too extreme: he was right about infinity but negative numbers could be constructed in a finite number of steps.
200 Weierstrass' expression of his attitude toward Kronecker
200 Kronecker's four foundational precepts of primitive intuitionism. I think these provide a good and proper foundation.
202 Kronecker's opposition to Cantor's work
203 Catholic theologians took up Cantor's cause after his defeat by Kronecker.
204 E.T. Bell's book discredited
205 The paradox and the correspondence do not occur if you consider a sequence of steps of construction. At each successive step you do not have all squares in the top row. Why would you conclude you do in the infinite limit?
205 Galileo, like Feynman, suspected something fishy about infinities.
205 Cantor's definition of an infinite set. I say it is vacuous.
206 Gauss: prior to Cantor: "...the use of infinite magnitude as if it were something finished; such use is not admissible in mathematics."
213 Hilbert's praise of Cantor's work: "the most admirable fruit of the mathematical mind and indeed one of the highest achievements of man's intellectual processes ... No one shall expel us from the paradise which Cantor has created for us."
214 Error. He means the "full set...is always a member of [its power set]"
214 The Continuum Hypothesis
214 Cantor's Paradox
215 Finitists (Aristotle, Gauss), Intuitionists (Kronecker, Brouwer), Transfinitists (Cantor, Hilbert)
222 "Foundations of Constructive Analysis", 1967 by Errett Bishop
222 Bishop: "Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself."
225 The Axiom of Choice
226 Goedel's Theorem
230 Solving the Four Color Problem
240 Cellular Automata and the "Game of Life"
243 Historical pattern of paradigms
244 Misguided (IMHO) notion of consciousness
245 Dawkins on memes
248 The universe may be discrete
249 Milton Shulman: "I knew a mathematician who said, 'I do not know as much as God, but I know as much as God did at my age.'"
251 The Greeks were first to develop an abstract notion of number.
251 The Pythagoreans saw numbers as the fundamental constituent of reality.
252 Philolaus of Croton (5th Century BC): "all things which can be known have number; for it is impossible for a thing to be conceived or known without number."
252 Barrow: "Beset by such beliefs, science as we know it could not develop. If all things were made of number, then the true nature of reality could only be discerned by studying numbers themselves, their interrelationships, their properties, divorced from any utilitarian motive." Yes! And Dr. Dick shows this to be the case!
252 Plato's ideals, or forms, defined the same set-theoretic way Prof. Grimm defined numbers. E.g. '3' is the set of all triples.
254 Aristotle's conclusion that ideal forms lead to infinite regress. I say let's figure out how to stop the regress at, say, 11.
255 Barrow seems to suggest considering Platonic forms to be imperfect! Hmmm. So do I!
255 Barrow: "Because the Platonic view points to innumerable particular representations of the same ideal formal concept, it places no significance upon those particular examples." Non sequitur! The "significance" is that they constitute the known world!
255 Platonic support for the theological notion of perfection and eternity.
255 Barrow's objection is the same as Aristotle's (p254). I suggest the same remedy.
256 I think Augustine should have considered the possibility that God is indeed constrained by logic.
256 These are not the only possible viewpoints. Consider mathematics to be like the game of chess. The player chooses to follow the rules and is thus bound by them. The player has the power and ability, nonetheless, to move a rook diagonally, but he cannot do so and remain consistent with the rules.
260 Hermite's objection to infinity as explained by Poincare
261 Goedel's acceptance of set-theoretic paradoxes
261 Penrose: the Mandelbrot Set is a discovery
264 Barrow: Goedel: There is only one set of "true" axioms.
264 Strange only because he believes reality itself to be complete. I don't.
268 Contrast between Arts and Sciences. Math seems to be Science.
268 I disagree with this paragraph. I think music and mathematics are exactly the same sort of thing in reality. Both are invented in THE mind, remembered, chosen and accepted on some chosen basis: consistency for math; predictability for music. Both consistency and predictability are pleasing. Both are incomplete, thus inconclusive. Both can be extended by going beyond the rules. In math, this results in new axioms, in music, new patterns or forms which thus become predictable and thus pleasing. They are both the same kind of structure available for discovery by human minds. The great composers had the music enter their human minds from outside.
272 Some advantages of Platonism
272 Barrow: "It is all very well to claim that ordinary objects are accounted for by another realm of abstract objects but why should any abstract realm have anything to do with any part of the mundane world." Me: If the creator of that abstract world is the creator of the physical world, then that would be the obvious result.
272 Barrow: "Where is this other world and how do we make contact with it?" Me: It is in the PC. We contact it by thinking.
272 Barrow: "How is it possible for our mind to have an interaction with the Platonic realm so that our brain state is altered by that experience?" Me: By a radio-like communication link.
273 Barrow: "...the Platonist must regard the best mathematicians as possessing a means of making contact with the Platonic world more often and more clearly than other individuals." Me: Yes. Same goes for musicians and poets.
273 Goethe: "What man does not know
Or has not thought of
Wanders in the night
Through the labyrinth of the mind."
274 My "PC" hypothesis answers all of Aaron Sloman's objections: "Where exactly is the Universe?" In the PC. "How fast does time really flow?" One second per second in this time dimension; not at all in others.
274 Barrow: "[W]hat is the source of mathematical intuition?" Me: Prior art by PC.
274 Barrow: "It is one thing to maintain that there exists another eternal Platonic world of mathematical forms but quite another to maintain that we can dip into it through some special mental effort." Me: We dip into it via the same conduit that provides consciousness and free will to brains.
274 Barrow: "The question of whether we have some particular type of mental connection with such a realm is one which seems to have been little considered by psychologists and neurophysiologists (presumably) because they do not take such a possibility very seriously." Me: Probably true or else they simply lack imagination.
276 Poincare: "A reality completely independent of the spirit that conceives it, sees it, or feels it, is an impossibility. A world so external as that, even if it existed, would be for ever inaccessible to us."
276 "arise"?, "by-product"? - Hardly persuasive.
276 Barrow: "There seems to be no way in which mathematical truths can interact with our minds to create mathematical knowledge of them." Me: Why not a radio-like communication link with PC?
277 Major assumption that individuals are different and distinct.
277 Sextus Empiricus: "...the abstract is not of a nature either to effect or to be affected." I say this is a false assumption; programs affect computer behavior.
278 Goedel suggests "another kind of relationship between ourselves and reality."
279 Penrose defending Platonism.
281 "simulation could give rise to...consciousness." GIANT unwarranted assertion which I strongly doubt. He uses this as the basis for much of his development in following pages which makes his conclusions groundless.
282 Gottlob Frege, Platonism, and proof of God's existence.
291 Groundless conclusion about emergent consciousness.
292 Emil Post on the implications of Goedel's Theorem.
293 Bertrand Russell's characterization of Philosophical Mysticism.
296 Where did he get that Platonism is "non-mental" or "timeless"?
Barrow's line of distinction separating mathematics into Platonism vs. consctuctionism could be improved. I think a better distinction is between finite and discrete mathematics vs. mathematics that include infinity. I think the acceptance of an infinite number of natural numbers is the necessary condition for Goedel's Theorem, not that it includes arithmetic. I think that Goedel's Theorem can't be proved in my proposed discrete system of Practical Numbers.
©2005 Paul R. Martin, All rights reserved.