Quote from Godfrey

But even more important is to understand what Epicurus was proposing!

Right -- I think the two go hand in hand.

A lot of this comes down to the struggle as to whether to consider the senses to be adequate to reveal reality to us, or whether we need something more (divine revelation, or the analogous "dialectical logic"). The Platonists and adherents of propositional logic want to consider the results of their calculations to transcend the reality of our senses, but in truth it doesn't, and is dependent on the reality of our senses to be relevant to us.,

Ultimately I think we can get a good glimpse by seeing what Lucretius is focusing on in the key section of Book 4, starting around line 470. Knowledge, including all valid rational analysis, is ultimately based on the senses.

So when you combine emphasis on the senses as our realm of reality and you layer on the passions (pleasure and pain) as the ultimate test of what "matters" to us then you've got a prescription for a full and complete approach to determining all truth that is relevant to us, Then when you add in the rejection of necessity (especially when it involves animate agency) you reject the gamesmanship involved in any kind of dialectical logic (which has nothing to do with any of those) because you always insist that the test of truth goes back (regardless of abstract formulas) to what we sense, pain/pleasure, and how we "anticipate."

Thanks to Kalosyni for this link! https://www.brainpickings.org/2014/01/03/bal…kit-carl-sagan/

Probably the most important part - two lists:

Nine Tools of Baloney Detection

1. Wherever possible there must be independent confirmation of the “facts.”
2. Encourage substantive debate on the evidence by knowledgeable proponents of all points of view.
3. Arguments from authority carry little weight — “authorities” have made mistakes in the past. They will do so again in the future. Perhaps a better way to say it is that in science there are no authorities; at most, there are experts.
4. Spin more than one hypothesis. If there’s something to be explained, think of all the different ways in which it could be explained. Then think of tests by which you might systematically disprove each of the alternatives. What survives, the hypothesis that resists disproof in this Darwinian selection among “multiple working hypotheses,” has a much better chance of being the right answer than if you had simply run with the first idea that caught your fancy.
5. Try not to get overly attached to a hypothesis just because it’s yours. It’s only a way station in the pursuit of knowledge. Ask yourself why you like the idea. Compare it fairly with the alternatives. See if you can find reasons for rejecting it. If you don’t, others will.
6. Quantify. If whatever it is you’re explaining has some measure, some numerical quantity attached to it, you’ll be much better able to discriminate among competing hypotheses. What is vague and qualitative is open to many explanations. Of course there are truths to be sought in the many qualitative issues we are obliged to confront, but finding them is more challenging.
7. If there’s a chain of argument, every link in the chain must work (including the premise) — not just most of them.
8. Occam’s Razor. This convenient rule-of-thumb urges us when faced with two hypotheses that explain the data equally well to choose the simpler.
9. Always ask whether the hypothesis can be, at least in principle, falsified. Propositions that are untestable, unfalsifiable are not worth much. Consider the grand idea that our Universe and everything in it is just an elementary particle — an electron, say — in a much bigger Cosmos. But if we can never acquire information from outside our Universe, is not the idea incapable of disproof? You must be able to check assertions out. Inveterate skeptics must be given the chance to follow your reasoning, to duplicate your experiments and see if they get the same result.

"Common Pitfalls of Common Sense"

1. ad hominem — Latin for “to the man,” attacking the arguer and not the argument (e.g., The Reverend Dr. Smith is a known Biblical fundamentalist, so her objections to evolution need not be taken seriously)
2. argument from authority (e.g., President Richard Nixon should be re-elected because he has a secret plan to end the war in Southeast Asia — but because it was secret, there was no way for the electorate to evaluate it on its merits; the argument amounted to trusting him because he was President: a mistake, as it turned out)
3. argument from adverse consequences (e.g., A God meting out punishment and reward must exist, because if He didn’t, society would be much more lawless and dangerous — perhaps even ungovernable. Or: The defendant in a widely publicized murder trial must be found guilty; otherwise, it will be an encouragement for other men to murder their wives)
4. appeal to ignorance — the claim that whatever has not been proved false must be true, and vice versa (e.g., There is no compelling evidence that UFOs are not visiting the Earth; therefore UFOs exist — and there is intelligent life elsewhere in the Universe. Or: There may be seventy kazillion other worlds, but not one is known to have the moral advancement of the Earth, so we’re still central to the Universe.) This impatience with ambiguity can be criticized in the phrase: absence of evidence is not evidence of absence.
5. special pleading, often to rescue a proposition in deep rhetorical trouble (e.g., How can a merciful God condemn future generations to torment because, against orders, one woman induced one man to eat an apple? Special plead: you don’t understand the subtle Doctrine of Free Will. Or: How can there be an equally godlike Father, Son, and Holy Ghost in the same Person? Special plead: You don’t understand the Divine Mystery of the Trinity. Or: How could God permit the followers of Judaism, Christianity, and Islam — each in their own way enjoined to heroic measures of loving kindness and compassion — to have perpetrated so much cruelty for so long? Special plead: You don’t understand Free Will again. And anyway, God moves in mysterious ways.)
6. begging the question, also called assuming the answer (e.g., We must institute the death penalty to discourage violent crime. But does the violent crime rate in fact fall when the death penalty is imposed? Or: The stock market fell yesterday because of a technical adjustment and profit-taking by investors — but is there any independent evidence for the causal role of “adjustment” and profit-taking; have we learned anything at all from this purported explanation?)
7. observational selection, also called the enumeration of favorable circumstances, or as the philosopher Francis Bacon described it, counting the hits and forgetting the misses (e.g., A state boasts of the Presidents it has produced, but is silent on its serial killers)
8. statistics of small numbers — a close relative of observational selection (e.g., “They say 1 out of every 5 people is Chinese. How is this possible? I know hundreds of people, and none of them is Chinese. Yours truly.” Or: “I’ve thrown three sevens in a row. Tonight I can’t lose.”)
9. misunderstanding of the nature of statistics (e.g., President Dwight Eisenhower expressing astonishment and alarm on discovering that fully half of all Americans have below average intelligence);
10. inconsistency (e.g., Prudently plan for the worst of which a potential military adversary is capable, but thriftily ignore scientific projections on environmental dangers because they’re not “proved.” Or: Attribute the declining life expectancy in the former Soviet Union to the failures of communism many years ago, but never attribute the high infant mortality rate in the United States (now highest of the major industrial nations) to the failures of capitalism. Or: Consider it reasonable for the Universe to continue to exist forever into the future, but judge absurd the possibility that it has infinite duration into the past);
11. non sequitur — Latin for “It doesn’t follow” (e.g., Our nation will prevail because God is great. But nearly every nation pretends this to be true; the German formulation was “Gott mit uns”). Often those falling into the non sequitur fallacy have simply failed to recognize alternative possibilities;
12. post hoc, ergo propter hoc — Latin for “It happened after, so it was caused by” (e.g., Jaime Cardinal Sin, Archbishop of Manila: “I know of … a 26-year-old who looks 60 because she takes [contraceptive] pills.” Or: Before women got the vote, there were no nuclear weapons)
13. meaningless question (e.g., What happens when an irresistible force meets an immovable object? But if there is such a thing as an irresistible force there can be no immovable objects, and vice versa)
14. excluded middle, or false dichotomy — considering only the two extremes in a continuum of intermediate possibilities (e.g., “Sure, take his side; my husband’s perfect; I’m always wrong.” Or: “Either you love your country or you hate it.” Or: “If you’re not part of the solution, you’re part of the problem”)
15. short-term vs. long-term — a subset of the excluded middle, but so important I’ve pulled it out for special attention (e.g., We can’t afford programs to feed malnourished children and educate pre-school kids. We need to urgently deal with crime on the streets. Or: Why explore space or pursue fundamental science when we have so huge a budget deficit?);
16. slippery slope, related to excluded middle (e.g., If we allow abortion in the first weeks of pregnancy, it will be impossible to prevent the killing of a full-term infant. Or, conversely: If the state prohibits abortion even in the ninth month, it will soon be telling us what to do with our bodies around the time of conception);
17. confusion of correlation and causation (e.g., A survey shows that more college graduates are homosexual than those with lesser education; therefore education makes people gay. Or: Andean earthquakes are correlated with closest approaches of the planet Uranus; therefore — despite the absence of any such correlation for the nearer, more massive planet Jupiter — the latter causes the former)
18. straw man — caricaturing a position to make it easier to attack (e.g., Scientists suppose that living things simply fell together by chance — a formulation that willfully ignores the central Darwinian insight, that Nature ratchets up by saving what works and discarding what doesn’t. Or — this is also a short-term/long-term fallacy — environmentalists care more for snail darters and spotted owls than they do for people)
19. suppressed evidence, or half-truths (e.g., An amazingly accurate and widely quoted “prophecy” of the assassination attempt on President Reagan is shown on television; but — an important detail — was it recorded before or after the event? Or: These government abuses demand revolution, even if you can’t make an omelette without breaking some eggs. Yes, but is this likely to be a revolution in which far more people are killed than under the previous regime? What does the experience of other revolutions suggest? Are all revolutions against oppressive regimes desirable and in the interests of the people?)
20. weasel words (e.g., The separation of powers of the U.S. Constitution specifies that the United States may not conduct a war without a declaration by Congress. On the other hand, Presidents are given control of foreign policy and the conduct of wars, which are potentially powerful tools for getting themselves re-elected. Presidents of either political party may therefore be tempted to arrange wars while waving the flag and calling the wars something else — “police actions,” “armed incursions,” “protective reaction strikes,” “pacification,” “safeguarding American interests,” and a wide variety of “operations,” such as “Operation Just Cause.” Euphemisms for war are one of a broad class of reinventions of language for political purposes. Talleyrand said, “An important art of politicians is to find new names for institutions which under old names have become odious to the public”)

That is extremely well written Nate thank you!

I particularly think your link between the Europeans and Emerson (transcendentalism) is helpful. It's been my experience (or at least viewpoint) that the word "transcendentalism" has too favorable a connotation at least in USA thought processes. They talk a lot about "nature" (Walden Pond?) and yet don't seem to be pointing to nature as much as they point to something that they allege "transends" nature.

It would be helpful to articulate in very clear terms to what extent it is fair to consider both them and the German idealists as essentially one with Platonic rejection of the "real world" in favor of the world of "ideas."

In regard to how this works with the transcendentalists I bet Joshua will have comment so I will tag him.

Again, not trying to oversimplify, but I always want us to drag these difficult ideas into the open at least in the form of "outlines" so that we can at least point the way to help people with a framework of understanding the issues and the differences with Epicurus.

Also need to note for Don't benefit here that while I still today think that the issue of absence of necessity, arising from human free will, is an important part of the refusal to say that hermarchus must be either alive or dead tomorrow, I continue last night's caveat that I could be wrong on that and that there may still be a purely logical point beyond necessity that Epicurus was concerned about.

So I would say that until that issue is resolved we're still on the hunt for the most exact way to express Epicurus' concern.

Maybe it's two steps that are independent of each other in the Hermarchus example ----

1 - there's no direct linkage (necessity) between the proposition and the conclusion. (a general objection to all propositional logic)

2 - the reason there's no direct linkage in this particular case is the presence of human free will. (the specific absence of linkage that applies in this case)

OMG that is very interesting! Thank you Joshua! What what a great Latin phrase for the pseudo-Romans like Don and me -- EX FALSO SEQUITUR QUODLIBET! How many occasions that fits!

And I bet you're right that if we researched Soissons we could find more that is relevant to the essential insight of the "It isn't necessary that Hermarchus be either alive or dead tomorrow so I'm not engaging in your game" observation!

(until such time as OMZ is established to mean Oh My Zeus I've stuck with OMG)

Now we have to know what a PARVIPTONIAN is!

A good time to restate the question:

What we are trying to do ultimately is get a firm fix on what it was that Epicurus was rejecting, while still embracing "reason" in PD16!

All this discussion of details is irrelevant and worthless unless we keep that goal in mind.

This also comes up and looks like it might be an interesting Paper, along the lines of the one Martin was quoting from in the presentation:

http://infolab.stanford.edu/~ullman/focs/ch12.pdf

Here's an article from the Internet Encyclopedia of Philosophy on "Propositional Logic" which appears to be becoming the term of choice to refer to what Epicurus questioned. Since many of the texts use "dialectic" however we probably still need to correlate those terms

Propositional Logic | Internet Encyclopedia of Philosophy

Propositional Logic

Propositional logic, also known as sentential logic and statement logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement. The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, which studies logical operators and connectives that are used to produce complex statements whose truth-value depends entirely on the truth-values of the simpler statements making them up, and in which it is assumed that every statement is either true or false and not both. However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another.

Or is it possible that we need to consider "Dialogical Logic"

Dialogical Logic | Internet Encyclopedia of Philosophy

Dialogical Logic

Dialogical logic is an approach to logic in which the meaning of the logical constants (connectives and quantifiers) and the notion of validity are explained in game-theoretic terms. The meaning of each logical constant (such as “and”, “or”, “implies”, “not”, “every”, and so forth) is given in terms of how assertions containing these logical constants can be attacked and defended in an adversarial dialogue. Dialogues are described as two-player games between a proponent and an opponent. A dialogue starts with an assertion made by the proponent. This assertion can then be attacked according to its logical form by the opponent. Depending upon the kind of attack, the proponent can now either defend against, or attack, the opponent’s move. The two players alternate until one player is unable to make another move. In this case, the dialogue is won by the other player who made the last move. An assertion made in the initial move by the proponent is said to be valid, if the proponent has a winning strategy for it, that is, if the proponent can win every dialogue for each possible move made by the opponent. The dialogical approach was initially worked out for intuitionistic logic and for classical logic; it has been extended to other logics, among them modal logic and linear logic.

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I see there does not appear to be an entry on "Dialectic"

Dialectic - Wikipedia

Dialectic

From Wikipedia, the free encyclopedia

Jump to navigationJump to searchFor varieties of language, see Dialect. For electrical insulators, see Dielectric.

Dialectic or dialectics (Greek: διαλεκτική, dialektikḗ; related to dialogue; German: Dialektik), also known as the dialectical method, is a discourse between two or more people holding different points of view about a subject but wishing to establish the truth through reasoned argumentation. Dialectic resembles debate, but the concept excludes subjective elements such as emotional appeal and the modern pejorative sense of rhetoric.^[1][2] Dialectic may thus be contrasted with both the eristic, which refers to argument that aims to successfully dispute another's argument (rather than searching for truth), and the didactic method, wherein one side of the conversation teaches the other. Dialectic is alternatively known as minor logic, as opposed to major logic or critique.

Within Hegelianism, the word dialectic has the specialised meaning of a contradiction between ideas that serves as the determining factor in their relationship. Dialectical materialism, a theory or set of theories produced mainly by Karl Marx and Friedrich Engels, adapted the Hegelian dialectic into arguments regarding traditional materialism. The dialectics of Hegel and Marx were criticized in the twentieth century by the philosophers Karl Popper and Mario Bunge.

Dialectic tends to imply a process of evolution and so does not naturally fit within classical logics, but was given some formalism in the twentieth century. The emphasis on process is particularly marked in Hegelian dialectic, and even more so in Marxist dialectical logic, which tried to account for the evolution of ideas over longer time periods in the real world.

of course THIS, referencing Popper, who is an author Martin has discussed reading:

Criticisms[edit]

Karl Popper has attacked the dialectic repeatedly. In 1937, he wrote and delivered a paper entitled "What Is Dialectic?" in which he attacked the dialectical method for its willingness "to put up with contradictions".^[62] Popper concluded the essay with these words: "The whole development of dialectic should be a warning against the dangers inherent in philosophical system-building. It should remind us that philosophy should not be made a basis for any sort of scientific system and that philosophers should be much more modest in their claims. One task which they can fulfill quite usefully is the study of the critical methods of science" (Ibid., p. 335).

In chapter 12 of volume 2 of The Open Society and Its Enemies (1944; 5th rev. ed., 1966), Popper unleashed a famous attack on Hegelian dialectics in which he held that Hegel's thought (unjustly in the view of some philosophers, such as Walter Kaufmann)^[63] was to some degree responsible for facilitating the rise of fascism in Europe by encouraging and justifying irrationalism. In section 17 of his 1961 "addenda" to The Open Society, entitled "Facts, Standards and Truth: A Further Criticism of Relativism", Popper refused to moderate his criticism of the Hegelian dialectic, arguing that it "played a major role in the downfall of the liberal movement in Germany [...] by contributing to historicism and to an identification of might and right, encouraged totalitarian modes of thought. [...] [And] undermined and eventually lowered the traditional standards of intellectual responsibility and honesty".^[64]

The philosopher of science and physicist Mario Bunge repeatedly criticized Hegelian and Marxian dialectics, calling them "fuzzy and remote from science"^[65] and a "disastrous legacy".^[66] He concluded: "The so-called laws of dialectics, such as formulated by Engels (1940, 1954) and Lenin (1947, 1981), are false insofar as they are intelligible."^[66]

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That last line is a good one: "False insofar as they are intelligible!"

Quote from Nate

I'd call German Idealism an extension or expansion of Plato, not a revival, but an addition

Yep that's my question. An addition of what?

Nate is there a way to summarize in even broader terms - maybe in Plato's own terms - how they were extending Platonic views?

when you say:

Quote from Nate

In general, they supported revelation over reason, faith over experience, and mind over matter.

I think that could be stated pretty exactly about Plato himself, right?

In what "general" ways were the German idealists seeing themselves as different?

Quote from Nate

postulates that we can never really know the "thing-in-itself", thus, faith and revelation become useful tools in a world that is completely mysterious.

Is that not also something that could be said of Plato too?

Although being well aware that in many cases things that are complex can't be reduced to simple terms, probably it's still helpful for generalist students of Epicurus to try to reduce them to "outline" form so we can at least get a handle on the categories to discuss.

Quote from Cassius

Would Epicurus reject German idealism less strongly, or more strongly, than he rejected Platonic idealism?

After reading through those Wikipedia notes it is clear that German idealism is strongly linked to Platonic idealism and thus it's probably very fair to say that Epicurus would reject both.

As yet I have no handle, however, on which of the two Epicurus would reject most strongly. My bet is that he would reject German idealism more strongly, if only on the grounds that after so much scientific knowledge had been added over thousands of years they still wanted to "double down" on Plato, but that's just a guess.

As Martin also observed last night in a phrase which I will quote from Jefferson's formulation "Nonsense can never be explained."

[Thomas Jefferson to John Adams, July 5, 1814: "The doctrines which flowed from the lips of Jesus himself are within the comprehension of a child; but thousands of volumes have not yet explained the Platonisms engrafted on them; and for this obvious reason, that nonsense can never be explained."]

German idealism

From Wikipedia, the free encyclopedia

The four principal German idealists: Immanuel Kant (upper left), Johann Gottlieb Fichte (upper right), Friedrich Wilhelm Joseph Schelling (lower left), Georg Wilhelm Friedrich Hegel (lower right)

German idealism was a philosophical movement that emerged in Germany in the late 18th and early 19th centuries. It developed out of the work of Immanuel Kant in the 1780s and 1790s,^[1] and was closely linked both with Romanticism and the revolutionary politics of the Enlightenment. The best-known thinkers in the movement, besides Kant, were Johann Gottlieb Fichte, Friedrich Wilhelm Joseph Schelling, Arthur Schopenhauer, Georg Wilhelm Friedrich Hegel, and the proponents of Jena Romanticism (Friedrich Hölderlin, Novalis, and Karl Wilhelm Friedrich Schlegel).^[2] August Ludwig Hülsen, Friedrich Heinrich Jacobi, Gottlob Ernst Schulze, Karl Leonhard Reinhold, Salomon Maimon and Friedrich Schleiermacher also made major contributions.

The period of German idealism after Kant is also known as post-Kantian idealism, post-Kantian philosophy, or simply post-Kantianism.^[3]

Fichte's philosophical work has controversially been interpreted as a stepping stone in the emergence of German speculative idealism, the thesis that we only ever have access to the correlation between thought and being.^[4] Another scheme divides German idealists into transcendental idealists, associated with Kant and Fichte, and absolute idealists, associated with Schelling and Hegel.

Meaning of idealism[edit]

Main article: Idealism

The word "idealism" has multiple meanings. The philosophical meaning of idealism are those properties we discover in objects that are dependent on the way that those objects appear to us, as perceived subjects. These properties only belong to the perceived appearance of the objects, and not something they possess "in themselves". The term "idea-ism" is closer to this intended meaning than the common notion of idealism. The question of what properties a thing might have "independently of the mind" is thus unknowable and a moot point, within the idealist tradition.

In our 20th discussion last night Martin mentioned Hegel and perhaps Schopenhauer, and I am beginning to realize I have a significant gap in my understanding of the place of what I gather is known as "German Idealism." I have in the past tended to think of that more in terms of some kind of "romanticism," but I gather there is much more of an issue as to its relationship to "Platonic idealism" that is worth being more clear about. I think I have a fair understanding of at least part of Nietzsche, and that probably he's more of a rebel against "German idealism" than a proponent of it, but at that point (and maybe not even at that point) I am lost.

So I'd like to start this thread about German Idealism so we can after some discussion perhaps come to some tentative thumbnail conclusions about how it relates to Epicurus.

At first glance, my working presumption is that if German idealism is a variant of Platonic idealism, then Epicurus would have very little good to say about it. Is even that a fair starting point for a tentative framework? Would Epicurus reject German idealism less strongly, or more strongly, than he rejected Platonic idealism?

Shall we continue for the sake of completeness?

PYTHAGORAS OF SAMOS

Biography – Who was Pythagoras

It is sometimes claimed that we owe pure mathematics to Pythagoras, and he is often called the first “true” mathematician. But, although his contribution was clearly important, he nevertheless remains a controversial figure.

He left no mathematical writings himself, and much of what we know about Pythagorean thought comes to us from the writings of Philolaus and other later Pythagorean scholars. Indeed, it is by no means clear whether many (or indeed any) of the theorems ascribed to him were in fact solved by Pythagoras personally or by his followers.

The school he established at Croton in southern Italy around 530 BCE was the nucleus of a rather bizarre Pythagorean sect. Although Pythagorean thought was largely dominated by mathematics, it was also profoundly mystical, and Pythagoras imposed his quasi-religious philosophies, strict vegetarianism, communal living, secret rites and odd rules on all the members of his school (including bizarre and apparently random edicts about never urinating towards the sun, never marrying a woman who wears gold jewellery, never passing an ass lying in the street, never eating or even touching black fava beans, etc) .

The members were divided into the “mathematikoi” (or “learners“), who extended and developed the more mathematical and scientific work that Pythagoras himself began, and the “akousmatikoi” (or “listeners“), who focused on the more religious and ritualistic aspects of his teachings. There was always a certain amount of friction between the two groups and eventually the sect became caught up in some fierce local fighting and ultimately dispersed. Resentment built up against the secrecy and exclusiveness of the Pythagoreans and, in 460 BCE, all their meeting places were burned and destroyed, with at least 50 members killed in Croton alone.

The over-riding dictum of Pythagoras’s school was “All is number” or “God is number”, and the Pythagoreans effectively practised a kind of numerology or number-worship, and considered each number to have its own character and meaning. For example, the number one was the generator of all numbers; two represented opinion; three, harmony; four, justice; five, marriage; six, creation; seven, the seven planets or “wandering stars”; etc. Odd numbers were thought of as female and even numbers as male.

The holiest number of all was “Tetractys” or ten, a triangular number composed of the sum of one, two, three and four. It is a great tribute to the Pythagoreans’ intellectual achievements that they deduced the special place of the number 10 from an abstract mathematical argument rather than from something as mundane as counting the fingers on two hands.

However, Pythagoras and his school – as well as a handful of other mathematicians of ancient Greece – was largely responsible for introducing a more rigorous mathematics than what had gone before, building from first principles using axioms and logic. Before Pythagoras, for example, geometry had been merely a collection of rules derived by empirical measurement.

Pythagoras discovered that a complete system of mathematics could be constructed, where geometric elements corresponded with numbers, and where integers and their ratios were all that was necessary to establish an entire system of logic and truth.

The Pythagorean Theorem

He is mainly remembered for what has become known as Pythagoras’ Theorem (or the Pythagorean Theorem): that, for any right-angled triangle, the square of the length of the hypotenuse (the longest side, opposite the right angle) is equal to the sum of the square of the other two sides (or “legs”).

Written as an equation: a² + b² = c².

What Pythagoras and his followers did not realize is that this also works for any shape: thus, the area of a pentagon on the hypotenuse is equal to the sum of the pentagons on the other two sides, as it does for a semi-circle or any other regular (or even irregular( shape.

The simplest and most commonly quoted example of a Pythagorean triangle is one with sides of 3, 4 and 5 units (3² + 4² = 5², as can be seen by drawing a grid of unit squares on each side as in the diagram at right), but there are a potentially infinite number of other integer “Pythagorean triples”, starting with (5, 12 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 40, 41), etc. It should be noted, however that (6, 8, 10) is not what is known as a “primitive” Pythagorean triple, because it is just a multiple of (3, 4, 5).

Pythagoras’ Theorem and the properties of right-angled triangles seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, and it was touched on in some of the most ancient mathematical texts from Babylon and Egypt, dating from over a thousand years earlier. One of the simplest proofs comes from ancient China, and probably dates from well before Pythagoras’ birth. It was Pythagoras, though, who gave the theorem its definitive form, although it is not clear whether Pythagoras himself definitively proved it or merely described it. Either way, it has become one of the best-known of all mathematical theorems, and as many as 400 different proofs now exist, some geometrical, some algebraic, some involving advanced differential equations, etc.

It soon became apparent, though, that non-integer solutions were also possible, so that an isosceles triangle with sides 1, 1 and √2, for example, also has a right angle, as the Babylonians had discovered centuries earlier. However, when Pythagoras’s student Hippasus tried to calculate the value of √2, he found that it was not possible to express it as a fraction, thereby indicating the potential existence of a whole new world of numbers, the irrational numbers (numbers that can not be expressed as simple fractions of integers). This discovery rather shattered the elegant mathematical world built up by Pythagoras and his followers, and the existence of a number that could not be expressed as the ratio of two of God’s creations (which is how they thought of the integers) jeopardized the cult’s entire belief system.

Poor Hippasus was apparently drowned by the secretive Pythagoreans for broadcasting this important discovery to the outside world. But the replacement of the idea of the divinity of the integers by the richer concept of the continuum, was an essential development in mathematics. It marked the real birth of Greek geometry, which deals with lines and planes and angles, all of which are continuous and not discrete.

Among his other achievements in geometry, Pythagoras (or at least his followers, the Pythagoreans) also realized that the sum of the angles of a triangle is equal to two right angles (180°), and probably also the generalization which states that the sum of the interior angles of a polygon with n sides is equal to (2n – 4) right angles, and that the sum of its exterior angles equals 4 right angles. They were able to construct figures of a given area, and to use simple geometrical algebra, for example to solve equations such as a(a – x) = x² by geometrical means.

The Pythagoreans also established the foundations of number theory, with their investigations of triangular, square and also perfect numbers (numbers that are the sum of their divisors). They discovered several new properties of square numbers, such as that the square of a number n is equal to the sum of the first n odd numbers (e.g. 4² = 16 = 1 + 3 + 5 + 7). They also discovered at least the first pair of amicable numbers, 220 and 284 (amicable numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220).

Music Theory

Pythagoras is also credited with the discovery that the intervals between harmonious musical notes always have whole number ratios. For instance, playing half a length of a guitar string gives the same note as the open string, but an octave higher; a third of a length gives a different but harmonious note; etc.

Non-whole number ratios, on the other hand, tend to give dissonant sounds. In this way, Pythagoras described the first four overtones which create the common intervals which have become the primary building blocks of musical harmony: the octave (1:1), the perfect fifth (3:2), the perfect fourth (4:3) and the major third (5:4). The oldest way of tuning the 12-note chromatic scale is known as Pythagorean tuning, and it is based on a stack of perfect fifths, each tuned in the ratio 3:2.

The mystical Pythagoras was so excited by this discovery that he became convinced that the whole universe was based on numbers, and that the planets and stars moved according to mathematical equations, which corresponded to musical notes, and thus produced a kind of symphony, the “Musical Universalis” or “Music of the Spheres”.

Accurate history or not, here is something also relevant, given that it is so widely accepted as true about Plato:

Plato FAQ: "Let no one ignorant of geometry enter"

"Let no one ignorant of geometry enter"

Tradition has it that this phrasewas engraved at the door of Plato's Academy, the school he had founded in Athens.

More detail: https://www.storyofmathematics.com/greek_plato.html

Biography: What was Plato Known for

Although usually remembered today as a philosopher, Plato was also one of ancient Greece’s most important patrons of mathematics.

Inspired by Pythagoras, he founded his Academy in Athens in 387 BCE, where he stressed mathematics as a way of understanding more about reality. In particular, he was convinced that geometry was the key to unlocking the secrets of the universe. The sign above the Academy entrance read: “Let no-one ignorant of geometry enter here”.

Plato played an important role in encouraging and inspiring Greek intellectuals to study mathematics as well as philosophy. His Academy taught mathematics as a branch of philosophy, as Pythagoras had done, and the first 10 years of the 15 year course at the Academy involved the study of science and mathematics, including plane and solid geometry, astronomy and harmonics. Plato became known as the “maker of mathematicians”, and his Academy boasted some of the most prominent mathematicians of the ancient world, including Eudoxus, Theaetetus and Archytas.

He demanded of his students accurate definitions, clearly stated assumptions, and logical deductive proof, and he insisted that geometric proofs be demonstrated with no aids other than a straight edge and a compass. Among the many mathematical problems Plato posed for his students’ investigation were the so-called Three Classical Problems (“squaring the circle”, “doubling the cube” and “trisecting the angle”) and to some extent these problems have become identified with Plato, although he was not the first to pose them.

Platonic Solids

Plato the mathematician is perhaps best known for his identification of 5 regular symmetrical 3-dimensional shapes, which he maintained were the basis for the whole universe, and which have become known as the Platonic Solids: the tetrahedron (constructed of 4 regular triangles, and which for Plato represented fire), the octahedron (composed of 8 triangles, representing air), the icosahedron (composed of 20 triangles, and representing water), the cube (composed of 6 squares, and representing earth), and the dodecahedron (made up of 12 pentagons, which Plato obscurely described as “the god used for arranging the constellations on the whole heaven”).

The tetrahedron, cube and dodecahedron were probably familiar to Pythagoras, and the octahedron and icosahedron were probably discovered by Theaetetus, a contemporary of Plato. Furthermore, it fell to Euclid, half a century later, to prove that these were the only possible convex regular polyhedra. But they nevertheless became popularly known as the Platonic Solids, and inspired mathematicians and geometers for many centuries to come. For example, around 1600, the German astronomer Johannes Kepler devised an ingenious system of nested Platonic solids and spheres to approximate quite well the distances of the known planets from the Sun (although he was enough of a scientist to abandon his elegant model when it proved to be not accurate enough).

One more from Lucian, similar to Hermotimus, this time from Incaromenippus, An Aerial Expedition. The main relevant part is in bold below but I left text before and after since it is so colorful:

Menippus. Well, a very short survey of life had convinced me of the absurdity and meanness and insecurity that pervade all human objects, such as wealth, office, power. I was filled with contempt for them, realized that to care for them was to lose all chance of what deserved care, and determined to grovel no more, but fix my gaze upon the great All. Here I found my first problem in what wise men call the universal order. I could not tell how it came into being, who made it, what was its beginning, or what its end. But my next step, which was the examination of details, landed me in yet worse perplexity. I found the stars dotted quite casually about the sky, and I wanted to know what the sun was. Especially the phenomena of the moon struck me as extraordinary, and quite passed my comprehension; there must be some mystery to account for those many phases, I conjectured. Nor could I feel any greater certainty about such things as the passage of lightning, the roll of thunder, the descent of rain and snow and hail.

In this state of mind, the best I could think of was to get at the truth of it all from the people called philosophers; they of course would be able to give it me. So I selected the best of them, if solemnity of visage, pallor of complexion and length of beard are any criterion—for there could not be a moment's doubt of their soaring words and heaven-high thoughts—and in their hands I placed myself. For a considerable sum down, and more to be paid when they should have perfected me in wisdom, I was to be made an airy metaphysician and instructed in the order of the universe. Unfortunately, so far from dispelling my previous ignorance, they perplexed me more and more, with their daily drenches of beginnings and ends, atoms and voids, matters and forms. My greatest difficulty was that, though they differed among themselves, and all they said was full of inconsistency and contradiction, they expected me to believe them, each pulling me in his own direction.

Friend. How absurd that wise men should quarrel about facts, and hold different opinions on the same things!

Menippus. Ah, but keep your laughter till you have heard something of their pretentious mystifications. To begin with, their feet are on the ground; they are no taller than the rest of us 'men that walk the earth'; they are no sharper-sighted than their neighbors, some of them purblind, indeed, with age or indolence. And yet they say they can distinguish the limits of the sky, they measure the sun's circumference, take their walks in the supra-lunar regions, and specify the sizes and shapes of the stars as though they had fallen from them. Often one of them could not tell you correctly the number of miles from Megara to Athens, but has no hesitation about the distance in feet from the sun to the moon. How high the atmosphere is, how deep the sea, how far it is round the earth— they have the figures for all that. Moreover, they have only to draw some circles, arrange a few triangles and squares, add certain complicated spheres, and lo, they have the cubic contents of Heaven.

Then, how reasonable and modest of them, dealing with subjects so debatable, to issue their views without a hint of uncertainty; thus it must be and it shall be; contra gentes they will have it so. They will tell you on oath the sun is a molten mass, the moon inhabited, and the stars water-drinkers, moisture being drawn up by the sun's rope and bucket and equitably distributed among them.

How their theories conflict is soon apparent; next-door neighbors? No, they are miles apart. In the first place, their views of the world differ. Some say it had no beginning, and cannot end; others boldly talk of its creator and his procedure. What particularly entertained me was that these latter set up a contriver of the universe, but fail to mention where he came from, or what he stood on while about his elaborate task, though it is by no means obvious how there could be place or time before the universe came into being.

Friend. You really do make them out very audacious conjurers.

Menippus. My dear fellow, I wish I could give you their lucubrations on ideas and incorporeals, on finite and infinite. Over that point, now, there is fierce battle; some circumscribe the All, others will have it unlimited. At the same time they declare for a plurality of worlds, and speak scornfully of others who make only one. And there is a bellicose person who maintains that war is the father of the universe.

As to Gods, I need hardly deal with that question. For some of them God is a number; some swear by dogs and geese and plane-trees. [note: Socrates made a practice of substituting these for the names of Gods in his oaths.] Some again banish all other Gods, and attribute the control of the universe to a single one; I got rather depressed on learning how small the supply of divinity was. But I was comforted by the lavish souls who not only make many, but classify; there was a First God, and second and third classes of divinity. Yet again, some regard the divine nature as unsubstantial and without form, while others conceive it as a substance. Then they were not all disposed to recognize a Providence; some relieve the Gods of all care, as we relieve the superannuated of their civic duties; in fact, they treat them exactly like supernumeraries on the stage. The last step is also taken, of saying that Gods do not exist at all, and leaving the world to drift along without a master or a guiding hand.

Well, when I heard all this, I dared not disbelieve people whose voices and beards were equally suggestive of Zeus. But I knew not where to turn for a theory that was not open to exception, nor combated by one as soon as propounded by another. I found myself in the state Homer has described; many a time I would vigorously start believing one of these gentlemen; “But then came second thoughts.”

So in my distress I began to despair of ever getting any knowledge about these things on earth. The only possible escape from perplexity would be to take to myself wings and go up to Heaven.

Also directly relevant to our topic tonight is Usener 376:

U376

Cicero Academica II.30.97 (Lucullus): They will not get Epicurus, who despises and laughs at the whole of dialectic, to admit the validity of a proposition of the form “Hermarchus will either be alive tomorrow or not alive,” while dialecticians demand that every disjunctive proposition of the form “either x or not-x” is not only valid but even necessary. See how on his guard the man is whom your friends think slow; for “If,” he says, “I admit either of the two to be necessary, it will follow that Hermarchus must either be alive tomorrow or not alive; but as a matter of fact in the nature of things no such necessity exists.” Therefore let the dialecticians, that is, Antiochus and the Stoics, do battle with this philosopher, for he overthrows the whole of dialectic.

Cicero, On The Nature of The Gods, I.25.70 (Cotta speaking): Epicurus did the same sort of thing in his argument with the logicians. It is an axiom of the traditional logic that in every disjunctive proposition of the form “X either is … or is not …” one of the alternatives must be true. He was afraid that if he admitted anything of this sort, then in a proposition such as “Tomorrow Epicurus will either be alive or he will not be alive,” one or the other of the statements would be a necessary truth: so to avoid this he denied that there was any logical necessity at all in a disjunction proposition, which is too stupid for words!

Cicero, On Fate, 10.21: Now here, first of all, if it were my desire to agree with Epicurus and deny that every proposition is either true or false, I would rather accept that blow than agree that all things come about through fate; for the former opinion gives some scope for discussion, but the latter is intolerable. So Chrysippus strains every sinew in order to convince us that every proposition is either true or false. Epicurus is afraid that, if he concedes this, he will have to concede that whatever comes about does so through fate; for if either the assertion or the denial is true from eternity, it will also be certain – and if certain, also necessary. [cf. Ibid., 9.19]

Potentially Relevant in Addition:

U380 (this may be a good clue to those categories in which dialectical formal logic is especially to be distrusted:

Aetius, Doxography, I.29.6 [p. 326 Diels] (Plutarch, I.29.2; Stobaeus Anthology, Physics 7.9): Epicurus says that chance is a cause which is uncertain with respect to persons, times, and places.

I would suggest that:

1 - Epicurus anticipated the need to say this by about 2000 years,

2 - That the need to say it was planted firmly at least as far back as Plato, and by every Greek philosopher before and after who rejected the senses as the ultimate source of all reliable source of information, and

3 - Orwell was right and in league with Epicurus in identifying lack of confidence in the senses as the ultimate weapon of those who consider themselves "the golden" elite:

Quote

The Party told you to reject the evidence of your eyes and ears. It was their final, most essential command. His heart sank as he thought of the enormous power arrayed against him, the ease with which any Party intellectual would overthrow him in debate, the subtle arguments which he would not be able to understand, much less answer. And yet he was in the right! They were wrong and he was right. The obvious, the silly, and the true had got to be defended. Truisms are true, hold on to that! The solid world exists, its laws do not change. Stones are hard, water is wet, objects unsupported fall toward the earth's center.

With the feeling that he was speaking to O'Brien, and also that he was setting forth an important axiom, he wrote:

Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.

George Orwell Nineteen Eight-Four - Book 1, Chapter 7.

(I know nothing about that page except that it came up first in my googling for a well-formatted copy.)

Episode Eighty-Eight of Lucretius Today is now available. In this episode we continue into Book Six to discuss the mysteries of the waters of the Nile and sulfur pits which are fatal to birds. Thanks to Don for reading this week:

In editing this episode I wanted to remember to include this line from Virgil which we discussed briefly but probably not nearly enough.

It's one of the great Latin lines with an Epicurean pedigree: