Let's Learn Philosophy, Theology and the History of Science

Tachyon · 2015-07-20 03:48:25

Tachyon · 2015-07-20 03:48:33

Introduction: The Great Throat-Clearing

TL;DR: Tachyon, who does not understand philosophy, reads about it and posts hopefully interesting stuff in semi-liveblog format. You are welcome to join in.

Longer explanation: I've been wanting to learn about philosophy for a while now, but before now I've never made a serious attempt to tackle the subject systematically. At present, I know extremely little about it, and I've never taken a class in it. I've taken a bachelor's course in English literature and creative writing, in which philosophy occasionally came up, and I've read Sophie's World. That's pretty much it. What little I do know concerns the Western tradition. What I know of Asian philosophy probably wouldn't fill a single side of A4, and I could write everything I know about philosophy from the rest of the world on a post-it note.

I decided to incorporate theology and the history of science because I believe they are so bound up with the history of philosophy that it's hard to extricate them. Actually, I'm going to be pretty broad in what I treat as philosophy, at least in part because I'm not 100% sure I know what philosophy is. There is every possibility I will spend a lot of time talking about mythology, legal texts, or even works of fiction that appear to be in some way relevant.

So anyway, why make this thread? Well, partly it's to motivate me to keep going with this, because I know I have a bad habit of starting projects and abandoning them. And partly it's because I think this is a subject probably a lot of people might find more interesting than they do but which can be pretty intimidating to beginners, so I thought maybe we could approach this together and learn stuff as we go along! If you're already very familiar with philosophy and/or science a lot of what I post here will probably be familiar to you, especially in the early posts. This will probably be particularly true if you know even the first thing about math, since math is something I often find quite daunting and probably do not understand so well as most people here. I'm hoping the gradual and comprehensive approach will pay off in the long run, the idea being to build on what we learn so that over time we end up with a more nuanced and holistic understanding of the ideas than if we rushed through it.

The ideal here would be to go through the entire tradition in roughly chronological order, which would probably take several lifetimes. I do plan on approaching the thinkers more or less 'in sequence', and situating them in their historical context as much as I can. There are two obvious limitations here, the first being that I'm doing this from home and don't have access to journals and stuff. I might buy books sometimes, but mostly I'm going to be reliant on the Internet. The second obstacle is that I can't read Greek, Chinese, Sanskrit, Latin, Pali, Hebrew, Arabic, Hindi, Italian or German. I have a B in GCSE French which may or may not be any use whatsoever. Mostly I'm going to be reliant on English translations where they exist, and these may not be online or in the public domain.

However, I'm going to start by reading an overview of the history of the discipline. I have a copy of Bertrand Russell's History of Western Philosophy, which is long and so far very readable, but from what I can gather also quite biased and outdated. I have also ordered a copy of Anthony Kenny's A New History of Western Philosophy, which seems to be well-regarded, albeit very dismissive of contemporary continental philosophy, and Victoria Harrison's Eastern Philosophy: The Basics, which I hope is a gentle but accurate introduction to Chinese and Indian philosophy. In addition, I'm (slowly) working my way through Peter Adamson's excellent podcast, A History of Philosophy Without Any Gaps. These are all broad works which I think will give me something of a foothold while I figure out my trajectory here; I request your patience during this time. I'm also armed with a bunch of dictionaries, plus anything I can find in my dad's rather large book collection that turns out to be relevant. And of course I'll be making use of Wikipedia, Gutenberg and any other relevant online resources I find.

If you've been following the Zestpoole thread you know how useless I am at keeping to any kind of schedule, but I'm going to try to update this at minimum once a week. I hope you'll join me and I hope this will be fun, or interesting, or hopefully both! Feedback, discussion, recommended reading and posts about your own studies all welcome and appreciated.

Tachyon · 2015-07-20 04:15:35

that took longer to type than i was expecting, because it turned out longer than i was expecting

but anyway

my first post here about actual philosophy will be some time this week, and will concern the pre-socratics

Klinotaxis · 2015-07-20 04:51:04

history of science outside of europe is going to be even harder to find than history of philosophy, if you weren't already aware. for china you have science and civilization, which is one of those books so comprehensive nobody's actually read it. for anywhere else, i have actually no clue, though i could find pointers for math in particular

Salix Babylonika · 2015-07-20 04:53:39

I feel like any book that begins publication in 1954 and ends it in 2004 is going to be not necessarily a super pop read.

Klinotaxis · 2015-07-20 05:31:39

not "not necessarily", but necessarily "necessarily not". that's ( modal) logic

Salix Babylonika · 2015-07-20 06:12:57

no I meant "not necessarily"

because you see, while it's unlikely, and thus not a certainty by any means, it is always technically possible that something like a multi-volume, deeply in-depth book on Chinese history and science could be a popular read.

whereas "necessarily not" would imply that I think such a thing to be literally impossible, which is certainly not true.

Tachyon · 2015-07-20 08:12:33

Oh i'm under no illusions that this is anything other than ridiculously huge undertaking. However much i do there'll still be loads more to look at. That's part of the fun i think.

Thanks for the recommendation! It looks rather more in-depth than i was planning to start out with, but i will keep it in mind for future reading. And yeah math stuff is good, if you know any resources about mathematical developments i'm unlikely to find in more conventional histories of the subject, i'd be interested. Particularly if you know of any primary sources that exist in translation or with commentaries.

i must admit, though, thusfar i've been envisioning the science stuff as a sort of side-dish, next to the philosophy main course. It may not stay this way, maybe this will turn out to be the start of a lifelong obsession with ancient Chinese astronomy, who knows?

Tachyon · 2015-07-20 08:23:22

Actually this brings up an issue i hadn't considered: the Western philosophical tradition pretty much started with natural philosophy, from the very beginning you see cosmology, geometry, even stuff about predicting the weather. i could be off-base here, but my impression is Chinese philosophy has its roots more in ethical and spiritual teachings, so probably doesn't intersect with developments in Chinese science so much, at least not during that very early period.

China was obviously the site of some of the most important developments in science and technology and to just skip over those would clearly be amiss. But it'll be interesting to see what and how much interaction there is between Chinese philosophy and Chinese developments in science.

Klinotaxis · 2015-07-20 14:30:59

Tachyon said:
i must admit, though, thusfar i've been envisioning the science stuff as a sort of side-dish, next to the philosophy main course. It may not stay this way, maybe this will turn out to be the start of a lifelong obsession with ancient Chinese astronomy, who knows?

Actually this brings up an issue i hadn't considered: the Western
philosophical tradition pretty much started with natural philosophy,
from the very beginning you see cosmology, geometry, even stuff about
predicting the weather. i could be off-base here, but my impression is
Chinese philosophy has its roots more in ethical and spiritual
teachings, so probably doesn't intersect with developments in Chinese
science so much, at least not during that very early period.

yeah, i know it's a side thing, i just brought that up because (a) i'm a science dude (b) i think there's really a lot of intersection with philosophy.

for instance the taoists/daoists/romanization sure sucks did a lot of alchemy; the "inner" alchemy is mystic religious-ish stuff that would be more at home in this thread, but there's also the "outer" alchemy which is, well, alchemy, chemicals, pouring mercury into your throat, that sorta shit. that's generally later than confucius tho.

Tachyon · 2015-07-20 18:50:10

Oh i think i heard of the mercury-swallowing thing! Wasn't there an emperor who died that way?

But yeah that does sound very interesting, and is a case in point of how little i know about daoist tradition.

Salix Babylonika · 2015-07-20 18:51:00

I think that was Sao Wong Di (that is a total guess I do not actually know how it's romanized)

same dude who is buried in that enormous tomb with the clay soldiers

Tachyon · 2015-07-20 18:56:56

Qin Shi Huang, sayeth Wikipedia. But yeah, of terracotta army fame.

Wikipedia said:
Reportedly, he died due to ingesting mercury pills, made by his alchemists and court physicians.^[74] Ironically, these pills were meant to make Qin Shi Huang immortal.^[74]

Salix Babylonika · 2015-07-20 18:58:17

Wow I really fucked that up.

Tachyon · 2015-07-20 18:59:28

You got the right guy though! Clay soldiers.

Calica · 2015-07-20 19:40:30

china and rome, though they had limited contact through the silk road, had a vague respect for each other as corresponding imperial powers. this apparently included dying of mercury poisoning

lee4hmz · 2015-07-20 19:56:08

wasn't mercury ascribed magical powers in ancient/medieval China?

Klinotaxis · 2015-07-20 21:19:51

it's cold liquid metal. anyone who saw it anywhere pretty much thought it was magic.

Calica · 2015-07-20 21:41:21

Calica said:
china and rome, though they had limited contact through the silk road, had a vague respect for each other as corresponding imperial powers. this apparently included dying of mercury poisoning

wait no for the roman emperors i was mostly thinking of lead.

still, metal poisoning

thenamelesssamurai · 2015-07-25 06:06:52

If you're interested, I have some philosophy of music texts from when I did a course on the subject that I can give to you. Though most of it deals with 20th century philosophy and music.

Tachyon · 2015-07-25 06:45:07

@thenamelesssamurai i'm interested. Are they in digital format?

i should note that it's going to be a while before i get to stuff that's recommended to me, at this stage, since i wanted to start out with stuff that's fairly old/basic before moving on to more complicated material and i'm not an especially fast reader. But if you don't mind that, then yes please.

thenamelesssamurai · 2015-07-25 06:57:23

Most of the stuff I have is in PDF format, except for the Dane Rudhyar stuff, which is available online. I can send you a zip file with all the pdfs tomorrow.

Here's the Rudyhar stuff: http://www.khaldea.com/rudhyar/music.html

Also, not related to philosophy, but if you're interested I can include Musimathics, which is a book about all the math involved in music, written in an approachable manner for people who haven't specialized in mathematics.

Tachyon · 2015-07-25 06:59:15

Please do! That sounds really interesting.

Thank you very much for this!

thenamelesssamurai · 2015-07-25 07:14:04

Tachyon said:
Please do! That sounds really interesting.

Thank you very much for this!

It's my pleasure. I'm really glad you're doing this topic. In a way, it's kinda reminiscent of a topic I've had kicking around in my head for 3-4 years, but never got around to making where people would teach each other philosophy by taking an essay or a book (or at lest a chapter of a book) they like and giving a casual, yet academic summary of the essay or book in question.

Tachyon · 2015-07-25 07:16:28

If you want to use this thread for that, feel free.

Or if you think it's something that would be better in its own thread, i'd read it.

thenamelesssamurai · 2015-07-25 07:17:56

Tachyon said:
If you want to use this thread for that, feel free.

Or if you think it's something that would be better in its own thread, i'd read it.

If I do it, I'd definitely want to keep it in its own thread and not clutter up your thread.

Tachyon · 2015-07-25 07:22:46

i don't think it would be clutter, but i appreciate the consideration.

i will have my first philosophy post up soon, hopefully this weekend. i'm thinking now that i might do a series of posts that are just very broad overviews of particular philosophical eras or schools, followed by a series on the major concerns of philosophy, like metaphysics and epistemology. Then i should like to take a look at some early philosophical or proto-philosophical texts.

thenamelesssamurai · 2015-07-25 07:26:16

Tachyon said:
i will have my first philosophy post up soon, hopefully this weekend. i'm thinking now that i might do a series of posts that are just very broad overviews of particular philosophical eras or schools, followed by a series on the major concerns of philosophy, like metaphysics and epistemology. Then i should like to take a look at some early philosophical or proto-philosophical texts.

I'd enjoy that style of introduction, considering most of the philosophy I know is only the most well known continental stuff.

Tachyon · 2015-08-03 17:25:33

Presocratic Philosophy: Introduction

Traditionally, Western philosophy begins with the ancient Greek thinkers known as the Presocratics, so called because they predate the death of Socrates in 399 BC. Obviously, the name was applied later. There was no single school of philosophy called the Presocratics, and depending on which book you consult they may not all have even been philosophers. Nevertheless they tend to be studied together and developed various ideas which would have an important influence on the philosophies of Plato, Aristotle and subsequent Greek philosophy.

The Presocratics are conventionally grouped into various schools (which may not have involved actual teacher-student relationships, although they are traditionally regarded as such), depending on where in the Mediterranean they lived and what their influences were. These include the Milesians (or Ionians), the Pythagoreans, the Eleatics and the Atomists. There are also some major figures that don't belong to any of these schools, such as Heraclitus and Empedocles.

With the notable exception of Democritus, who was a contemporary of Socrates, very little of the work of these thinkers survives today. Most of what we know comes from fragments, or from the writings of Aristotle and Theophrastus.

Tachyon · 2015-08-03 17:28:14

The Milesians: A Brief Overview

Presocratic philosophy begins in Miletus, on the western coast of Ionia, in present day Turkey. The Milesians were very interested in nature and produced a series of speculative cosmologies. Unfortunately, very little of their writing has survived, so we don't know whether they arrived at their views through philosophical argument, nor do we know, if so, what their arguments might have been.

The first of the Milesian philosophers, and of the Presocratics, was Thales of Miletus, who lived in the early 6th century BC. Tradition states that he learned geometry in Egypt, and credits him with the theorem that an angle inscribed in a semicircle is a right angle. It is not known how he proved this, but according to Diogenes Laertius he sacrificed an ox to give thanks for his discovery. He calculated the heights of the pyramids by measuring their shadows, and he applied his knowledge of geometry to calculate the distance of ships at sea. He is also credited with predicting a solar eclipse in 585 BC, but this has to have involved some lucky guesswork since the ancients had no way of predicting where a solar eclipse would be visible; furthermore this prediction could only have been made with reference to Babylonian astronomy tables and probably involved no independent work on Thales' part.

Thales is best known for claiming that everything is made of water. Obviously this, like a lot of the ideas in ancient philosophy, sounds kind of daft from a modern perspective, but let's take it seriously. We need to be wary of anachronism; clearly, Thales was not making the self-refuting claim that the most fundamental thing is H₂O. Everything being made of water would mean that rocks, clouds, trees and human beings are all made of the same stuff - so why single out water?

Thales must have thought water was in some way more basic or more important than other things, but we don't know how he arrived at this view. Perhaps he observed the water cycle and thought that if water could become ice and cloud, then why not stone and flesh? Perhaps he was extrapolating from semen being moist, or from living things needing water to survive. We don't know. In Thales' cosmology, the world is flat, and floats on the ocean like a log. Thales also apparently believed that everything is full of gods. We don't know what he meant by this, or why he believed it.

There are two anecdotes associated with Thales, which might be taken to suggest an interest in climatology and astronomy, respectively, but which we might doubt since they are clearly both stories told to make a point. The more flattering of the two is the story of how Thales, predicting a good olive harvest, bought up all the local olive presses one year and made a small fortune, thereby showing that his wisdom had practical applications and that he could be rich if he wanted to be. The less flattering anecdote, conversely, portrays him as a man with his head in the clouds, who fell into a hole because he was watching the stars instead of where he was going.

According to tradition, Anaximander was Thales' student, and Anaximenes was Anaximander's student. Anaximander's cosmology was in some respects similar to that of Thales, but in other respects very different. Like Thales, he believed that everything was formed from a single stuff, but rather than identify this with any earthly substance, he called it the 'apeiron', meaning 'boundless' or 'unlimited'. The apeiron, according to Anaximander, was everlasting and infinitely big, and was the source of everything in the universe. Various pairs of opposites were separated out from this apeiron, the first pair being hot and cold. Sometimes one opposite would dominate the other, but would pay a penalty and render reparation for this, so over time no one opposite would overtake the others. By this process of justice, the universe was formed. Less poetically, we might take this to mean that definite things were generated from the indefinite apeiron, and developed into a universe through a process of continuous opposition to one another. Entire worlds were separated out of the apeiron, although it's not clear whether Anaximander meant to suggest that our world is just one of many, or to posit a kind of cycle of successive worlds being created and destroyed.

In Anaximander's cosmology, the Earth is a squat cylinder. It is located at the very centre of the universe, meaning that there is no reason for it to be pulled one way any more than any other; this was supposed to explain its apparent lack of movement. In this cosmology, the world is surrounded by mist, and beyond that, enormous fiery hoops that encircle the world. These were once a vast ball of fire which surrounded the Earth, but broke apart into circles. These hoops then developed a barklike coating, but holes in the coating allow us to glimpse the fires inside; these are the heavenly bodies. However, sometimes a blockage will form in one of the hoops, and the result is an eclipse.

Anaximander does not seem to have entirely turned his back on Thales' idea, that everything comes from water. He believed that the world was slowly drying out, as the oceans gradually evaporated and became wind. He seems to have been postulating a time in the past when the world was much wetter than it is now, since he claimed that the earliest creatures formed in moisture. Observing that small children couldn't survive without their parents, he reasoned that there must have been a time when human beings were not the same way that they are now. He speculated that at some time in the past, humans looked like fish, or perhaps grew inside fish, and burst out of their scales upon reaching maturity. For this reason, he advised against eating fish.

Anaximander seems to have been well-travelled, and was credited with producing a map of the world. He also introduced the Greeks to the sundial, which he may have learned from the Babylonians. He also took an interest in astronomy, and produced a star chart, the first of the Greeks to do so.

Anaximenes was the third and final of the Milesians, and the last great philosopher active in Ionia before the destruction of Miletus by the Persians in 494 BC. Anaximenes identified Anaximander's indefinite apeiron with the air, and produced a cosmology similar to Thales' but with air taking the place of water. According to Anaximenes, stable air is invisible. When air becomes more rarefied, it becomes hotter and becomes fire, while as it becomes denser it becomes colder, becoming liquid, and eventually solid matter like earth. Anaximander purported to be able to prove, empirically, that air temperature was linked to density, by an experiment that you can try out yourself. Hold your hand in front of your face and blow on your hand twice, first with your lips pursed and the second time with your mouth wide open. The second time should feel warmer.

Air was then an infinite substance, surrounding the earth from every direction. The Earth was a disc of thickened air, floating on the thinner air like a leaf. He likewise compared the heavenly bodies to fiery leaves, also floating on the air. They rotate around the Earth like a felt hat being spun around somebody's head. The Earth tilts as it floats, and this is why not all the heavenly bodies are visible all the time.

Anaximenes also identified the soul with air, in the form of breath, or pneuma. Thus air is the source of all living things, including the gods. And of course, when things stop breathing, they die.

And that's about it for the Milesians! Below i've listed the references i used when compiling this and the previous post, for if you want to follow along with me or explore any of these ideas in more depth.

Peter Adamson, A History of Philosophy Without Any Gaps, Everything is Full of Gods: Thales, Infinity and Beyond: Anaximander and Anaximenes

Simon Blackburn, The Oxford Dictionary of Philosophy, revised second edition (Oxford: Oxford University Press, 2008) 978-0-19-954143-0

Will Buckingham, Douglas Burnham, Clive Hill, Peter J. King, John Marenbon and Marcus Weeks, The Philosophy Book (London: Dorling Kindersley, 2011) 978-1-4093-7053-6

Christopher Clapham and James Nicholson, The Concise Oxford Dictionary of Mathematics, fourth edition (Oxford: Oxford University Press, 2009) 978-0-19-923594-0

Anthony Kenny, A New History of Western Philosophy (Oxford: Oxford University Press, 2012) 978-0-19-965649-3

Bertrand Russell, History of Western Philosophy (Abingdon: Routledge, 2004) 978-0-415-32505-9

Tachyon · 2015-08-03 17:42:22

So . . . yeah! If anyone would like to offer feedback or corrections, please go ahead, since this stuff is all quite new to me.

my next post will be about the Pythagoreans. i've done a little reading on all the major Presocratic thinkers but i need some time to structure it into a post.

The current plan is that from there i'll follow the tradition of philosophy in Europe, Southwest Asia and North Africa up until the time of colonialism. Then i'll track back and follow the traditions of philosophy in India and China before tracing the history of philosophy up to the present, including postcolonial theory and the major concerns in today's analytic and continental philosophy. i apologize in advance for the slightly Eurocentric bias this entails; i've found it a lot easier to come by resources on Western philosophy and i want to be able to do the Eastern traditions justice, which means i'd rather hold off on them until i've had a bit more time to go over the material.

Also if anyone has any recommendations for introductory texts on the history of philosophy in India, or on other less well-known traditions like that of sub-Saharan Africa, please let me know.

glennmagusharvey · 2015-08-03 19:07:58

> so we don't know whether they arrived at their views through philosophical argument

What's the significance of "philosophical argument", or producing a philosophical viewpoint or understanding through this, as opposed to other means?

Alternatively, what does that term mean?

> Thales also apparently believed that everything is full of gods. We don't know what he meant by this, or why he believed it.

This is also believed by various religious and cultural traditions (or was, for some that are no longer practiced).

This, and the stuff that you're describing of Anaximander and the "apeiron", illustrates how religion, origin stories, the sciences, and even philosophical understanding all stem from various attempts of people to gain a conceptual understanding of how stuff works around them.

Tachyon · 2015-08-03 21:43:22

Indeed.

When i say 'philosophical argument', i suppose i really just mean 'argument'. That is, we don't know if the Milesians used logic and debate to arrive at their conclusions, the techniques we would normally regard as philosophical. Anthony Kenny, a former President of the British Academy and the Royal Institute of Philosophy, argues that the Milesians weren't really philosophers since they didn't engage in conceptual analysis. Nor were they scientists, because they didn't carry out rigorous experiments. What they did was offer speculative cosmologies which moved away from, but didn't attempt to overturn, the existing Greek mythology of the period.

To answer your question more comprehensively i'd need to have a firm definition of what philosophy *is*. As Kenny puts it, the word means different things in different mouths. To the 20th century Welsh philosopher Bertrand Russell, for instance, it refers to an intermediate field between science and theology, a speculative area which is neither definite knowledge or religious dogma. For myself, i think i'd prefer to hold off on giving a definition, at least until i've spent a little more time familiarizing myself with the thinkers we call 'philosophers' in English.

Your Thales point is a good one. The problem is that his statement is ambiguous, at least as it comes down to us in the present (bearing in mind that Thales was already somewhat distant history in Aristotle's day!). Thales apparently believed that magnets were living things, since they're capable of movement, apparently of their own accord. Perhaps he was thinking along those lines.

Tachyon · 2015-08-03 22:11:02

Per @Crystal's request, i've updated the op with amazon links to the books cited.

glennmagusharvey · 2015-08-03 22:11:59

fucking magnets, how do they work

Anyway I guess it makes sense if you're saying the distinction of what is or isn't philosophy is whether someone is just dreaming up postulating some sort of meaning/interpretation of the world around them, as opposed to supposing that and subsequently challenging it with logic or inference or something more rigorous along those lines.

Tachyon · 2015-08-03 22:18:10

That seems as good a working definition as any.

thenamelesssamurai · 2015-08-05 00:34:29

Alright, sorry this took so long. Here is the link: http://www.mediafire.com/download/kiud11fsxs8j7vc/Philosophy_of_music.rar

The contents are as follows (for anyone interested):
Main readings from the class:
Harry Partch - Genesis of a Music
Jacques Attali - Noise: The Political Economy of Music
Dane Rudyhar - The Magic of Tone and the Art of Music (previously linked)
Luigi Russolo - Art of Noises
John Cage - Silence: lectures and writings by John Cage
Adorno – Philosophy of New Music
Adrono – Essays in Music
Stockhausen – Toward a Cosmic Music (the last two are not included because I couldn't find them in PDF form, you'll have to find a physical copy. Sorry.)

Supplemental reading:
Derek Bailey - Improvisation: Its Nature And Practice In Music
Iannis Xenakis - Formalized Music: Thought and Mathematics in Composition
Adorno Aesthetic Theory
Douglas Hofstadter - Gödel, Escher, Bach
Anything archived on the Dane Rudyhar archives
Alex Ross - The Rest is Noise (not really a philosophy text, but it is a nice history of 20th century music and provides some nice context for some of these writings. Also a good book in general.)
John Cage - John Cage: Writer: Selected Texts
Iannis Xenakis - Arts/Sciences: Alloys (Aesthetics in Music)
Pierre Schæffer - In Search of a Concrete Music (last 3 are not included because I couldn't find them in PDF form)

Supplemental reading not related to my class:
Everything from Yarrun's class on experimental music, available here: http://www.mediafire.com/download/1bnl5ugm1f7lubh/Experimental+Stuff.rar

Not related to philosophy but still interesting:
Musimathics Vol 1 & 2

Tachyon · 2015-08-05 00:46:35

Thanks @thenamelesssamurai!

Tachyon · 2015-08-05 00:58:50

holy crap this is the entire GEB isn't it

i actually have this book, it's huge

thenamelesssamurai · 2015-08-05 01:36:56

Tachyon said:
holy crap this is the entire GEB isn't it

i actually have this book, it's huge

Yes it is and yeah it is a monster of a book.

Tachyon · 2015-08-09 21:00:22

The Pythagoreans: A Brief Overview

Mathematics predates philosophy. The Egyptians, for example, had a fairly sophisticated mathematics which was mainly used for practical purposes such as accounting and land surveying (a "geometer" was originally a person who measured land). The Babylonians, too, used mathematics for a variety of practical purposes, but also in their astronomy, and for a long time astronomy and even astrology were regarded as branches of mathematics. However, the development of pure mathematics was a gradual one, and the philosophy of Pythagoras was an important influence.

Even if you don't know the first thing about philosophy, you've heard of Pythagoras. You know that the square of the hypotenuse is the sum of the squares of the other two sides. But if you don't know about his philosophy, you might be surprised to learn that the historical Pythagoras was a mystic, worshipped by some as a god.

Pythagoras was born in Samos, off the coast of Ionia. He seems to have been well-travelled, and possibly visited Miletus and Egypt. At the age of 40 he moved to Croton, in Southern Italy, then inhabited by Greek colonists, where he started a kind of religious community of about 300 people, known as the Pythagoreans. Pythagoras was banished from Croton following a revolution in 510 BC and died in Metapontum at the end of the century. His followers continued to practice Pythagoreanism at Croton until they were scattered in 450 BC.

The details of Pythagoras' life and teachings are hazy, in part because the Pythagoreans observed a code of silence and didn't teach their discoveries to outsiders. They also had a tendency to attribute their ideas and discoveries to Pythagoras himself, which makes it difficult to discern what the historical Pythagoras really believed. They followed a number of dietary rules, and did not eat beans or certain meats. They also observed various other taboos; for example, a Pythagorean was prohibited from touching a white cockerel, or from stirring the fire with iron. Women were deemed more pious than men, a view derived from the Greek Bacchic religion.

The Pythagoreans taught that "all is number" - that is, they believed the entire universe to be governed by rational numbers (numbers that can be expressed as a fraction, such as 1/2 or 1/1). While the discovery of irrational numbers proved this to be false, the belief led them to study mathematics for its own sake, and to apply it in novel contexts such as music. The Pythagoreans associated musical scales with ratios. Under this model, if 1/1 is a note, 2/1 is an octave, 5/4 is a major third and 3/2 is a fifth. They also believed that the heavenly bodies were distributed according to similar ratios.

For the Pythagoreans, the discovery of a mathematical proof was a kind of religious ecstasy. While mathematics itself had existed for a long time, the Pythagorean emphasis on proofs was new, and distinguishes Greek mathematics from the earlier mathematics of Babylon and Egypt. The Egyptians knew that a triangle with sides of length 3, 4, and 5 must have a right angle, and the Babylonians had observed, empirically, that the square of the hypotenuse was equal to the sum of the squares on the other two sides, but they don't seem to have felt the need to prove it.

Pythagorean number theory also incorporated a great deal of mysticism, however. For example, 2 represented woman, 3 represented man, 2+3=5, so 5 represented marriage. A more enduring idea was the use of 1 to represent a point, 2 to represent a line, 3 a plane and 4 a solid. They particularly liked the number 10, being the sum of the first 4 integers, and so posited that there must be 10 heavenly bodies. For this reason, the Pythagorean cosmology includes a 'counter-Earth' hidden from view. Another tendency of the Pythagoreans was to categorize numbers by the shapes made by arrangements of pebbles; numbers of the series (1, 3, 6, 10...) were called triangular numbers, and numbers of the series (1, 4, 9, 16...) were square numbers.

The most important doctrine of Pythagoreanism was metempsychosis, the transmigration of the soul. This was the beginning of a long tradition of dualism in philosophy. The idea that the soul survives the death of the body was not new, and was borrowed from the Greek Orphic religion. Rather than a shadowy afterlife, however, Pythagoras taught that the soul enters a new body after death, and so is reborn as a new human or animal. Pythagoras even claimed to remember events from his own past lives. Some Pythagoreans believed in a cosmic cycle of rebirth: after a person died, his or her soul would be reborn into every kind of animal over the course of 3000 years before being reborn as a human once again. However, they believed that when Pythagoras died, he was reborn as a god. It's not difficult to see a parallel between the Pythagoreans' belief in the immaterial, eternal soul and their interest in immaterial, eternally true mathematics.

Notable among the Pythagoreans was Philolaus, from Croton, the first Greek thinker known to have regarded the Earth as a planet. It is to Philolaus that we owe most of what we know about the Pythagorean cosmology.

Another notable Pythagorean was Hippasus of Metapontum, credited with the discovery of irrational numbers. The discovery specifically concerned the square root of two, and was achieved using the Pythagorean theorem. The argument goes as follows: assume a right-angled triangle with 2 sides of unit length. Per the Pythagorean theorem, the hypotenuse must be equal to the square root of two. According to Pythagorean doctrine, the square root of two would have to be a rational number m/n such that (m/n)²=2. If we write this fraction in its simplest terms, either m or n must be odd (otherwise we could keep simplifying it). If we then multiply both sides of this equation by n², we get m²=2n², meaning that m must be an even number, which we'll call 2p, and n must be odd. 2²=4 so we can write our equation 4p²=2n². If we then divide both sides by 2 we get 2p²=n², making n² even, meaning that n must also be even. But as we've already noted, n can't be even, so we have a contradiction. Therefore there are no integers m and n that satisfy (m/n)²=2, so the square root of 2 must be irrational. Hippasus was allegedly punished harshly for this violation of Pythagorean doctrine, possibly by drowning.

Pythagoreanism persisted long after the disbanding of the Pythagorean community. There were Pythagoreans at Plato's Academy, and both Plato and Aristotle were influenced by Pythagorean teachings. There was a resurgence of Pythagoreanism in the 1st century BC, and again between the 3rd and 6th centuries AD, when the traditional religion came under threat from the spread of Christianity. Pythagoras became a kind of pagan answer to Jesus, and was claimed by Platonists such as Porphyry and Iamblichus to be the son of the God Apollo, and a worker of miracles. Around AD 100, the Pythagorean Nicomachus wrote a book, Introductio Arithmetica, notable for its break with the Euclidean tradition of representing numbers as geometrical quantities (such as lengths or areas). Nicomachus treated numbers as purely abstract, as in modern number theory - but he only worked with rational numbers.

References used:

Peter Adamson, A History of Philosophy Without Any Gaps, The Man With The Golden Thigh: Pythagoras, Serafina Cuomo on Ancient Mathematics (with Serafina Cuomo)

Simon Blackburn, The Oxford Dictionary of Philosophy, revised second edition (Oxford: Oxford University Press, 2008) 978-0-19-954143-0

Will Buckingham, Douglas Burnham, Clive Hill, Peter J. King, John Marenbon and Marcus Weeks, The Philosophy Book (London: Dorling Kindersley, 2011) 978-1-4093-7053-6

Tony Crilly, The Big Questions: Mathematics (London: Quercus, 2011) 978-1-84916-240-1

Anthony Kenny, A New History of Western Philosophy (Oxford: Oxford University Press, 2012) 978-0-19-965649-3

Bertrand Russell, History of Western Philosophy (Abingdon: Routledge, 2004) 978-0-415-32505-9

Ian Stewart, Why Beauty Is Truth: A History of Symmetry (New York, NY: Basic Books, 2007) 978-0-465-08237-7

Tachyon · 2015-08-10 00:48:20

BTW mathposts are probably the ones where i'm most liable to make stupid mistakes, so please do correct me if i do.

The next post will be up hopefully some time later this week, and will concern religion in Ancient Greece, and a somewhat lesser known thinker, Xenophanes.

Odradek · 2015-08-10 00:50:35

Who I will never stop confusing with Xenophon til the day I die

Tachyon · 2015-08-10 00:52:57

On that note, it's quite confusing that there are 2 major Greek philosophers called Zeno, and 3 Presocratics whose names start with Anax-.

Klinotaxis · 2015-08-10 00:55:32

you could be as dumb as me and confuse with xenocrates

Odradek · 2015-08-10 00:58:26

Tachyon said:
On that note, it's quite confusing that there are 2 major Greek philosophers called Zeno, and 3 Presocratics whose names start with Anax-.

It was a time before surnames

Klinotaxis · 2015-08-10 01:00:50

Tachyon said:
The Pythagoreans associated musical scales with ratios. Under this model, if 1/1 is a note, 2/1 is an octave, 5/4 is a major third and 3/2 is a fifth. They also believed that the heavenly bodies were distributed according to similar ratios.

the way you wrote this implies that they were wrong, but this is in fact how music worked (and works, sometimes). if you play two pitches apart by an octave on, say, a string instrument, the frequency of vibration will actually be in a 2/1 ratio (or the vibrating string will be half as long, is how these peeps would have understood it).

heavenly bodies aren't usually "distributed according to ratios" but do sometimes have orbital periods that are related to one another by simple ratios; couldn't tell you if the pythagoreans had empirical basis for that shit.

Tachyon · 2015-08-10 01:04:33

Poor phrasing on my part; i didn't actually mean to suggest that they were mistaken, only that the association of pitches with ratios in Greece was a Pythagorean innovation.

As far as i can tell the heavenly bodies thing had more to do with mysticism than empirical observation (hence the postulating an unseen 10th heavenly body purely on the basis that 10 is the "best" number). Cool, though, about the orbital resonance.

Klinotaxis · 2015-08-10 01:07:40

analyzing frequencies as such might have been an innovation, but i'm reasonably sure that people were playing fifths before math. you can just sort of hear it.

as for orbital resonances that ties into this whole thing about how once you have periodic waves you start running into small integers. works for quantum too. wack-y i tells ya.

Tachyon · 2015-08-10 01:12:01

Point taken, on the music.

Like with all the Presocratics, any information about Pythagoras should be treated with some amount of scepticism. Iamblichus, for instance, credits Pythagoras not only with the invention of the scale but with the invention of a whole host of musical instruments, but given that he also apparently believed Pythagoras was a god who could talk to animals and see the future, we should probably not take that too seriously.

glennmagusharvey · 2015-08-10 02:03:44

Thanks for the write-up.

> Samos

Is there something that makes a lot of these Greek islands have names that end in "-os"?

> 510 BC, 450 BC

Suddenly I find myself desiring a year numbering system that has no negative numbers. Unfortunately, documented human history doesn't quite stretch far back enough...

> did not eat beans or certain meats

weirdos

> prohibited from touching a white cockerel

weirdos

> the discovery of a mathematical proof was a kind of religious ecstasy

If only this were the case for today's schoolchildren.

> 2 represented woman, 3 represented man, 2+3=5, so 5 represented marriage.

weirdos

> Some Pythagoreans believed in a cosmic cycle of rebirth: after a person died, his or her soul would be reborn into every kind of animal over the course of 3000 years before being reborn as a human once again.

What do the Hare Krishna folks say about this duration? For some reason I am thinking 300,000 years but maybe that was the duration of the longest-period world layer or something. I remember hearing this from them on one occasion.

> Hippasus was allegedly punished harshly for this violation of Pythagorean doctrine, possibly by drowning.

weirdo assholes

> Nichomachus treated numbers as purely abstract, as in modern number theory - but he only worked with rational numbers.

To be fair even today we don't exactly have a good way of representing irrational numbers.

> credits Pythagoras not only with the invention of the scale but with the invention of a whole host of musical instruments

Hey I remember that bit in Donald Duck in Mathmagic Land.

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