Andrew Dynneson October 2015

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1 Fractionalization of Mathematics. Many of the mathematicians that we have discussed thus far, including Plato, Pythagoras, Euclid, Aristotle, Descartes, Newton, Leibniz, Hilbert, Russel, etc, are also Philosophers, and are natural philosophers, in the sense that they were also physicists, and concerned themselves with a mathematical theory of physical phenomena. Today these areas of research are strongly divided. It is often the case that Physicists and Mathematicians seldom speak to one another, and the same goes for those that study Philosophy. There is a subtle feeling of co-intimidation that lies just beneath the surface. A Physics expert will be intimidated by an expert in Mathematics, and vice versa. There is sometimes a breakdown in communication between three departments. Even Physics and Chemistry are somewhat at-odds. Mathematics itself is divided into sub-fields and specialties. There are Analysts, Topologists, Algebraists, Geometers, Logicians, Statisticians, Applied Mathematicians. There are even Mathematical Physicists (we sometimes joke that mathematical physicists are alienated from both mathematicians and physicists simultaneously). One asks when and how did this division occur? When did Mathematics/Philosophy become fragmented and compartmentalized? Kant: Critique of Pure Reason (c.1780s). Obviously it is difficult to point to a single event in history as being the fractionalization of mathematics. However, many would agree that the publication of Kant's Critique of Pure Reason marked a decided shift in Western thinking. This book points out that it may not be possible to distinguish between the perception of things, and the things themselves. The former is called phenomena, that which is attained by the senses. The noemena, the actual objects or events, are merely hypothesized by perceiving that which is perceived. How do I know that this projector I refer to as projector really exists on its own, or is it merely sense phenomena? This brings to mind modern works such as Bertrand Russel's The Problems of Philosophy (1912), which poses similar question of existence and non-existence. In effect, one cannot distinguish between the perception of eating a chip, the taste and texture, and the existence of the chip itself. These questions also bring to mind the earlier mention of Descartes's Meditations on First Philosophy (1639), it which Descartes concludes that this only thing he knows to truly exist is his own self. If you accept his reasoning, then by proxy, the only thing you accept to truly exist is your own self. However someone else (who may or may not exist)

convinced you of this conclusion (haha). At any rate, I am digressing at this point. After Kant, science became an investigation of phenomena, using empirical methods.

org/philosophy/when-and-how-did-science-become-separatedfrom-philosophy-t18854.html ) Hence, the split between Science and Philosophy. Copernican Revolution.

2 convinced you of this conclusion (haha). At any rate, I am digressing at this point. After Kant, science became an investigation of phenomena, using empirical methods. Meanwhile, the questions of the interplay between perceived and the actual became relegated to questions of metaphysics. ( ) Hence, the split between Science and Philosophy. Copernican Revolution. Nicolaus Copernicus, a Polish Astronomer/Mathematician who receives the historical credit for positing the Heliocentric (Sun at the center) model of the Universe. It was in 1543 that his book De Revolutionibus Orbium Coelestium laid the groundwork for what Kant referred to as the Copernican Revolution of Philosophy. [There is some mention that the Pythagoreans believed that the Earth was spherical, and also not at the center, but that it rotated about some unseen flame (Fraser, Craig G. (2006).The Cosmos: A Historical Perspective. p.14.).] The proponents of the Geocentric Model (Earth in the Center) performed some admirable feats, performing finer and finer corrections on the model, trying to force the perception of the Universe revolving about the Earth into the mathematics. Among the better known, we have Ptolemy (c.1550) Here is a snapshot of the motions of Venus and the Sun, whilest holding the Earth as fixed. I am not sure how accurate this is, since I pulled this from a questionable internet-site: Illustration 1: Wooden Books Illustration 2: Here we have another image, showing the trajectories of various planets, holding the earth as Fixed. rogram.com/threads/ Making-Earth-the- Center-of-the-Solar- System However, the point is clear. The motion of these planets about the Earth is at once Chaotic and

3 aesthetic. However, by observing the Copernican model, observe how the motions become much simplified: It is almost as though Heliocentrism is forced upon us by the observation of phenomena, the motion of heavenly bodies. Clearly, the center is difficult to prove, however the trajectories are much more consistent when we suppose the Sun to be at the center than that of the Earth. Illustration 3: magegallery/heliocentricsmall.jpg Hardyism. Once again, I stress that it is difficult to point to an exact event as the root cause of these divisions, however once can view certain historical events as major contributors to the split, indeed they were in the minds of some people as being the outright-cause. The Manhattan Project was actually instigated my Einstein and another physicist by the name of Szilard in 1939, and the project made use of several thousands of scientists ( ). The rational may have been that since these physicists had themselves fled Nazi Germany, it was logical to assume that Germany had also realized that the Theory of Relativity could be used to create a weapon of unimaginable destruction. It then became necessary to build the atomic bomb before Germany finished building theirs, so that Germany could be wiped from the face of the earth before the rest of the free world. The equation that is often spoken-of in hushed tones is E=mc^2, essentially this equation says that mass is equivalent to Energy, and that a small amount of mass equals a very large amount of energy, since it is multiplied by the speed of light times the speed of light again. The speed of light is a very large quantity, since it travels very quickly. An estimate for the amount of Energy released during the first atomic test in Almogordo New Mexico in 1945 is conservative to 10^14 Joules (kg m^2/s^2) (See Estimate of the energy released in the first Atomic Bomb explosion):

These scientists took a theory of unimaginable beauty, and created a weapon of unimaginable horror, the fear of which would consume the minds of a generation, and continues to affect us today.

4 These scientists took a theory of unimaginable beauty, and created a weapon of unimaginable horror, the fear of which would consume the minds of a generation, and continues to affect us today. Although Nazi Germany was defeated before America was able to build and test an atomic bomb, the threat of Japan became an opportunity for President Truman to utilize this ultimate weapon. Nuclear weapons were deployed to destroy the Japanese cities of Hiroshima and Nagasaki in This is marked as one of the most horrifying events in history, putting the death-toll from ground-zero as well as lingering nuclear fall-out effects at about 340,000. This ended the war between the US and Japan, however the consequences remained. The proliferation of nuclear weapons proceeded, as world powers felt that they must possess as many atomic weapons as possible, or perish utterly. So many weapons were built, that if launched, the destruction of the entire planet was eminent. The lead physicist on the Manhattan Project was Robert Oppenheimer, himself familiar with Eastern religions, in particular the Bhagavad Gita, remarked, I am become death, the destroyer of worlds. Einstein remarked in his letter, On My Participation in the Atom Bomb Project: I was well aware of the dreadful danger for all mankind, if these experiments would succeed. But the probability that the Germans might work on that very problem with good chance of success prompted me to take that step. I did not see any other way out, although I always was a convinced pacifist. To kill in war time, it seems to me, is in no ways better than common murder. Hardyism. It was during this time that many mathematicians began to experience a growing feeling of dejection and remorse that their discoveries had led in whole or in part to one of the most horrible creations in history. It would have been a simple matter to feel at least somewhat responsible for the deaths of those Japanese lives. Even if we did not directly push-the-button, our knowledge was used to put the button into-commission. In particular, this brings to mind the sentiment of G.H. Hardy ( ): "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."

This marked a shift in the thinking of mathematicians at the time. There arose a new movement to perform pure mathematics, which is opposed to applied mathematics.

5 This marked a shift in the thinking of mathematicians at the time. There arose a new movement to perform pure mathematics, which is opposed to applied mathematics. Pure math need not have an application. For mathematicians like Hardy who partook of this sentiment, they made a point of hoping that their discoveries would not have any applications at-all. Thus for some, Math became a sort of art, rather than a science. Here is where I trace the split between Science and Mathematics. Math is in-fact the language of the sciences, however it has also taken a life of its own, as a sort of mosaic. You can now perform mathematical research in an area that has no known applications to the everyday world. Many times, I have noticed that the original intentions and purposes of mathematical studies are absolutely forgotten, and the subjects are re-purposed to be more math for the sake of math, and utility in math becomes how useful a theorem is at proving other theorems. See Davis and Hersh: The Mathematical Experience (1981). Conclusion. Thus we have a three-fold split: Math, Physics and Philosophy are no longer the unifiedfront that it once was. This lecture is not to be interpreted as a plea for reunification of the fields. If I were to describe my own sentiment, it is rather to be interpreted as an awareness of these divisions, and an opportunity to explore inter-disciplinary communication, with our common roots in-mind. Perhaps we can even begin to heal those rifts that have occurred historically.

Bell s spaceship paradox

Bell s spaceship paradox If the two ships start accelerating at the same time, I always see them travelling at the same velocity, and keeping a constant distance... But I said the objects get shorter when