קאווע שטיבל

With respect to being, Meinong states that all complete objects have either being or non-being, and all incomplete objects lack being. Meinong’s indifference principle accommodates the law of excluded middle in the external but not in the internal version. From the fact that incomplete objects do not have being, it does not follow that they have non-being as well. Some of them, for example, incomplete objects that are also contradictorily determined, have non-being; others, like the triangle as such, are not determined with respect to being at all

Meinong stressed the distinction between “being existing” as a determination of so-being and “existence” (“to exist”) as a determination of being. Meinong’s distinction between judgements of so-being and judgements of being, combined with the indifference principle that being does not belong to the object’s nature (so-being), reminds one of Kant’s dictum that being is not a real predicate. Meinong did not accept the ontological argument either, and argued that “being existing” is a determination of so-being and can in a certain sense be properly accepted even of the object “existing golden mountain,” and, say, even of the object “existing round square,” whereas “existence”, which is a determination of being, will no more belong to the one than it does to the other. In other words, according to the characterization principle both the existing round square and the existing golden mountain (Russell 1907 uses “existent”) are existing (existent), but nevertheless they do not exist. Bertrand Russell could not see any difference between “being existent” and “to exist” and had “no more to say on this head.”

Being is that which belongs to every conceivable term, to every possible object of thought... “A is not” must always be either false or meaningless. For if A were nothing, it could not be said not to be... Numbers, the Homeric gods, relations, chimeras, and four-dimensional spaces all have being, for if they were not entities of a kind, we could make no propositions about them. Thus being is a general attribute of everything, and to mention anything is to show that it is. Existence, on the contrary, is the prerogative of some only amongst beings

I’m moved to laughter at the thought of how presumptous it would be to reject mathematics for philosophical reasons. How would you like the job of telling the mathematicians that they must change their ways, and abjure countless errors, now that philosophy has discovered that there are no classes? Can you tell them, with a straight face, to follow philosophical argument wherever it may lead? If they challenge your credentials, will you boast of philosophy’s other great discoveries: that motion is impossible, that a Being than which no greater can be conceived cannot be conceived not to exist, that it is unthinkable that anything exists outside the mind, that time is unreal, that no theory has ever been made at all probable by evidence (but on the other hand that an empirically adequate ideal theory cannot possibly be false), that it is a wide-open scientific question whether anyone has ever believed anything, and so on, and on, ad nauseum? Not me!

That may suggest!an ‘anything goes’ attitude toward philosophical questions that I neither hold nor approve of. I would insist that when debate over a philosophical question ends in deadlock, it does not follow that there is no truth of the matter; or that we don’t know the truth of the matter; or that we ought to suspend judgement; or that we have no reason for thinking one thing rather than the other

מי אני האט געשריבן:אז מ׳האט אויבן גערעדט אין די געדאנק פון דאָבּל-נעגעישאן לגבי מגדיר זיין ״ג-ט״, איז כידוע דא צוויי סארטן דראָגס: אינהיבּטארי, וואס האלטן צוריק ניוראָטראסמיטארס אינעם מח, און עקסייטעטארי, וואס טוהן דאס פארקערטע. אלקאהאל איז אַן אינהיבּיטארי דראָג וואס (ווי דר. פּאָל בּלום איז דאס מסביר אין די לעקציע [27:06-27:15]) טוהט אינהיבּיטן די אינהיבּיטאָרי חלקים פונעם מח (הגם באמת זענען די חלקים אין ראם פון עקסייטעטארי) וואס האלטן אין אלגעמיין צוריק דעם מענטש, ולכן איז א שיכור ווילדער וכדומה.

ולפי״ז טרעפט מען אזא סארט ענליכן געדאנק פון א דאָבּל-נעגאטיוו ביי יין. האב איך געקלערט אז מ׳קען מיט דעם זאגן א רמז בהגמרא בעירובין סה. א״ר חנינא כל המתפתה ביינו יש בו מדעת קונו ע״ש. והיינו, אז ער פארשטייט די געדאנק פונעם דאָבּל-נעגאטיוו שבהיין, וואס די געדאנק איז לגבי לדעת את קונו וככל הנ״ל. (און דאס קען אויך זיין די רמז דארט פון נכנס יין יצא סוד. און דאס קען אויך זיין די געדאנק פון, סנהדרין ע., חמרא וריחני פקחין.)

מי אני האט געשריבן:דר. פרעד סאממערס פארציילט אז אלס תלמיד פון הגאון הרב דר. סאלאווייטשיק האט ער אים געזאגט, ״איך האב ספיקות איבער דאס מציאות פון ג-ט״. האט ער ארומגעלייגט זיין האנט ארום אים און געזאגט, ״פרעד, אינטעלעקטועלי איז עס 50/50. מ׳דארף גלייבן״.

I have no confidence whatsoever in his claim that in order to believe in God I need to have a probability significantly higher than 51%. He gave no argument for that other than just an example. He gave an illustration that he thinks it is 51% probable that Hillary Clinton will win the presidential election. But that is not enough for him to believe that she will win. Well, fine, that is just an illustration. But in many other cases, 51% might be enough for a person to believe in something, especially (as William James pointed out) if this is a pressing conclusion that imposes itself upon you and is a matter of great urgency and importance, William James argued that in a case like that you are perfectly rational to believe in something even without evidence. This is James' famous essay, “The Will to Believe.” This idea that in order to believe something you have to be significantly more than 51% confident in it, I think is person-relative and subjective. For some people that might be true, for other people it might not. For some beliefs it might be true, but for other beliefs (as James said) it may not. I agree – I have no confidence in his argument from confidence!

Dr. Scharp's seems to think that the probability of the conclusion of a deductive argument cannot be any higher than the probability of the conjunction of the premises. You take the premises together and ask, “How probable is it that all of these premises are true?” And that will give you the probability of the conclusion. That is false in a deductive argument. The probability of the conjunction of the premises only sets a minimum probability for the conclusion. So if the probability of the premises is 51%, all that proves is that the probability of the conclusion cannot be less than 51%

It shows merely that the probability of the conclusion cannot be less than 51%. But it could be much higher. It could be 70, 80, 90%. I think Dr. Scharp was under the misimpression that the conclusion to a deductive argument has a probability which is simply equal to the probability of the conjunction of the premises

Allen Wood adduces significant textual evidence which confirms his emphasis upon Kant's conviction that belief in God's existence cannot be morally required, that is, there cannot be a moral obligation to believe that God exists (nor can belief in his existence be justly required by the state). Kant held these convictions out of respect for freedom of conscience, a disdain for hypocrisy, and because they fitted with his belief that the best 'theoretical' evidence and arguments could only establish God's logical possibility, that the concept of God is not self-contradictory. As Wood correctly notes, Kant did suggest that the 'minimum of theology' everyone can be expected to accept is the minimum compatible with this theoretical evidence, that is, that 'it is possible that there is a God', and sometimes claimed that 'religion' does not necessarily involve 'assertoric faith' in God, that is, belief in the existence of God, as opposed to his mere possibility. Thus, it is indeed possible for an agnostic (what Kant liked to call a 'sceptical' as opposed to a 'dogmatic' atheist) to be religious in Kant's sense

Yet, from the claim that religion may not necessarily involve assertoric faith in the existence of God, it does not follow that assertoric faith cannot be rationally required. In fact, Kant consistently asserted that it is. In his lectures on religion in the 1780s, for example, Kant compared the epistemic status of the 'firm conviction' in the postulate of God's existence with that of mathematical axioms

Gödel starts with an axiom—an assumption, in other words: If φ has the property P and from φ always follows ψ, then ψ also has the property P. For simplicity, we can assume that P stands for “positive”. For example:, if a fruit is delicious, a positive property, then it is also fun to eat. Therefore, the fun of eating it is also a positive property

The second axiom further sets a framework for P. If the opposite of something is positive, then that “something” must be negative. Thus, Gödel has divided a world into black and white: Either something is good or bad. For example, if health is good, then a disease must necessarily be bad

With these two premises, Gödel can derive his first theorem: If φ is a positive property, then there is a possibility that an x with property φ exists. That is, it is possible for positive things to exist

Now the mathematician turns for the first time to the definition of a divine being: x is divine if it possesses all positive properties φ. The second axiom ensures that a God defined in this way cannot have negative characteristics (otherwise one would create a contradiction)

The third axiom states that divinity is a positive characteristic. This point is not really arguable because divinity combines all positive characteristics

The second theorem now becomes a bit more concrete: by combining the third axiom (divinity is positive) and the first theorem (there is the possibility that something positive exists), a being x could exist that is divine

Gödel’s goal now is to show in the following steps that God must necessarily exist in the framework that has been laid out. For this purpose, he introduces in the second definition the “essence” φ of an object x, a characteristic property that determines all other characteristics. An illustrative example is “puppylike if something has this property, it is necessarily cute, fluffy and clumsy

The fourth axiom doesn’t seem too exciting at first. It simply states that if something is positive, then it is always positive — no matter the time, situation or place. Being puppylike and tasting good, for example, are always positive, whether during the day or at night in Heidelberg, Germany, or Buenos Aires

Gödel can now formulate the third theorem: if a being x is divine, then divinity is its essential property. This makes sense because if something is divine, it possesses all positive characteristics — and thus the properties of x are fixed

The next step relates to the existence of a particular being. If somewhere at least one being y possesses the property φ, which is the essential property of x, then x also exists. That is, if anything is puppylike, then puppies must also exist

According to the fifth axiom, existence is a positive property. I think most people would agree with that

From this one can now conclude that God exists because this being possesses every positive property, and existence is positive

As it turns out, Gödel’s logical inferences are all correct — even computers have been able to prove that. Nevertheless, these inferences have also drawn criticism. Besides the axioms, which of course can be questioned (why should a world be divisible into “good” and “evil”?), Gödel does not give more details about what a positive property is

It’s true that by means of the definitions and axioms, one can describe the set P mathematically:

1. If a property belongs to the set, its negation is not included. The set is self-contained

2. The fact that the essence of the set has only the characteristics of the set is itself an element of the set. The set always has the same elements — independent of the situation. In this case, the situation is the mathematical model in which the set is contained

3. Existence is part of the set

4. If φ is part of the set, then the property of having φ as the essence of the set is also contained in the set

But all of this does not ensure that this set is unique. There could be multiple collections that satisfy the requirements. For example, as logicians have shown, it is possible to construct cases where, by Gödel’s definition, there are more than 700 divine entities that differ in essence

Maimonides’ appeal to negation (GP 1.58) is often misunderstood because in normal speech a double negative usually indicates a positive. If I say that this dog is not lacking in the power of sight, you would be justified in concluding that it can see for the simple reason that sight is a power normally associated with dogs. What Maimonides has in mind is a more extreme form of negation. Thus “God is powerful” means “God does not lack power or possess it in a way that makes it comparable to other things.” Can God do something like move a book off a shelf? Yes, to the extent that God does not lack power but no to the extent that God does not have to move muscles, summon energy, or receive a supply of food or fuel. The power to create the whole universe is so far beyond that needed to move a book that any comparison cannot help but mislead

הקטן האט געשריבן:ס'איז היינט אן אפענע וועלט און ווער ס'וויל לערנען חכמה האט ב"ה אוואו צו גיין.

These are dangerous times. Never have so many people had access to so much knowledge, and yet been so resistant to learning anything

What is most thought-provoking in these thought-provoking times, is that we are still not thinking

מי אני האט געשריבן:עס קומט אויס אז לגבי דאס פּראפּאזישאן פון דאס ״עקזיסטענץ״ פון ג-ט איז עס כעין 1 - 1 [איינס מיינוס איינס; פּאזיטיוו 1 און דערנאך צוגעלייגט צו דעם נעגאטיוו 1], און דער אטעאיסט זאגט דאך מעיקרא אויף דעם א גאנצע 0. איז די שאלה דא צי ביי די פאל איז 1-1 ≠ 0 אדער יא 1-1 = 0 ווי ביי געהעריגע מאטעמאטיקס? מיינענדיג, איז עס טאקע די זעלבע ווי דער אטעאיסט (עכ״פ בעלמא הדין) צי נישט?

Integration [of God into mathematics] is not possible in mathematics. In mathematics, God’s revelation is silent. There is nothing to integrate… as far as math goes, there ain’t nuthin’ there

מי אני האט געשריבן:אז מ׳האט דערמאנט גראַנדי׳ס סיריִס בנוגע רעליגיע האב איך געקלערט ענליך אז דא [פון 1:50] איז מען מסביר, עפ״י דר. אנטאני פלוּ, אז ס׳איז יתכן לומר אז דער אטעאיסט און דער טעאיסט זאגן בעצם די זעלבע זאך אויף למעשה (עכ״פ לגבי די אביעקטיווע וועלט ממש). דאס איז נאך מער עפ״י א שטארקע מהלך פון טעאיסטישע נעגאטיווע טעאלאגיע. און ווי אונז האבן דא אראפגעברענגט:

מי אני האט געשריבן:דאס דערמאנט פון דעם פראנצויזישן טעאלאג מישעל דע סערטעא'ס מימרא:

The weight of [God's] transcendence makes any proposition relative, even to the point that the statement "God exists" has to be followed by a denial

והיינו, אז עס קומט אויס אז לגבי דאס פּראפּאזישאן אן פון דאס ״עקזיסטענץ״ פון ג-ט איז עס כעין 1 - 1 [איינס מיינוס איינס; פּאזיטיוו 1 און דערנאך צוגעלייגט צו דעם נעגאטיוו 1], און דער אטעאיסט זאגט דאך מעיקרא אויף דעם א גאנצע 0. איז די שאלה דא צי ביי די פאל איז 1-1 ≠ 0 אדער יא 1-1 = 0 ווי ביי געהעריגע מאטעמאטיקס? מיינענדיג, איז עס טאקע די זעלבע ווי דער אטעאיסט (עכ״פ בעלמא הדין) צי נישט?

‏(דאס קען שוין אויך אָנרירן אויף די געדאנק.)

דאס געבט אפשר א נייעם באדייט/קנייטש צו יוּקליד׳ס מימרא:

The laws of nature are but the mathematical thoughts of G-d

ווי אויך צו דר. מעקס טעגמארק׳ס היפאטעזיע…

A second denial of theological truth comes from the quarter of Eastern mysticism and its peculiarly Western step-child, the New Age movement. According to this perspective, which I shall call mystical Anti-Realism, there are propositions about God all right, but they are neither true nor false; they are all of them truth valueless. Thus, propositions expressed by sentences like “God exists,” “God is good,” or “The world was created by God” are neither true nor false, having no truth value. God is said to transcend all categories of human thought and language, so that it is quite impossible to assert any truths about God, as Christian theology pretends to do

Unfortunately, it’s not even clear what is meant by the Mystical Anti-Realist claim that God is “above human thought and language.” This is a metaphorical expression; but what does it mean? The best sense I can make of this claim is that what logicians call the Principle of Bivalence fails to be valid for propositions about God. The Principle of Bivalence states that for any proposition p, p is either true or false. The Principle is very closely related to the Law of Excluded Middle, one of the famous three “laws of thoughts,” which states that for any proposition p and its negation not-p, either p is true or not-p is true. The claim under consideration is that propositions ostensibly referring to God are neither true nor false

Now on the face of it such a position seems incomprehensible, for it seems absurd to say that a logical contradiction is not false. But on this view a proposition expressed by a sentence like “God both exists and does not exist” is not false. Such a proposition seems necessarily false! Neither is it true that “God either exists or does not exist.” But this statement seems to be necessarily true — what other alternative is there?

But the position involves an even deeper incoherence. For consider the proposition expressed by the sentence: “God can be described by bivalent propositions.” Since that proposition is itself a proposition about God, the Principle of Bivalence should not be valid for it. Therefore, it cannot have a truth value; in particular, it cannot be false. But if it is not false, then how can it be the case, as the Anti-Realist claims, that the Principle of Bivalence fails for propositions about God? If the Principle of Bivalence fails for propositions about God, then isn’t it false that God can be described by bivalent propositions? The claim thus refutes itself: one cannot coherently affirm that propositions about God are neither true nor false

The Anti-Realist might retort that the above only shows that rational paradox is inevitable when we try to talk about God

מי אני האט געשריבן:‫איך האב געקלערט בדרך צחות אז לגבי ווי אזוי איך האב צוגעשטעלט דעם געדאנק פון אמונה צו גראַנדי׳ס סיריִס אז מ׳קען אין דעם אריינטייטשן דעם אויבן-דערמאנטן טאמסאן׳ס לאמפ חקירה. דאס שטעלט צאם גראַנדי׳ס סיריִס מיט די סיריִס פון 1+1/2+1/4+1/8 אא״וו וועלכעס קאנווערדזשד צו 2, אזוי ווי מ׳האט דערמאנט אנהויב אשכול. והיינו, די חקירה פרעגט אז אויב איך האב א לאמפ וואס איך צינד דאס אן נאכ׳ן דורכגיין 1 מינוט [לגבי די גראַנדי סיריִס וועט דאס האבן א וועליוּ פון 1] און איך לעש עס דערנאך אויס נאכ׳ן דורכגיין 1/2 מינוט [לגבי די גראַנדי סיריִס וועט דאס האבן א וועליוּ פון (1-)], דערנאך צינד איך עס צוריק אָן נאכ׳ן דורכגיין 1/4 מינוט, דערנאך לעש איך עס אויס נאכ׳ן דורכגיין 1/8 מינוט, איך צינד עס נאכאמאל אָן נאכ׳ן דורכגיין 1/16 מינוט, וכן הלאה והלאה. ווי דערמאנט גייען די אלע אינטערוואלס, הגם זיי זענען אינפיניט, קאנווערדזשען צו 2 [מינוט]. שטעלט זיך די שאלה, ביי 2 מינוט איז די לאמפ אנגעצונדען אדער אויסגעלאשן? דאז ווענדט זיך אינעם גראַנדי סיריִס וויבאלד בתוך די 2 מינוט איז דאך דא אן אינפיניט צאל פון געהאלבטע אינטערוואלס, וואס ביי יעדע פון זיי טוישט זיך עס אלטערנאטיוולי פון (1-) [אויסגעלאשן] צו 1 [אנגעצינדען], און די סיריִס קאנווערדזשד דאך נישט געהעריג כנ״ל. און אן ענטפער פון 1/2 ביי דעם אין די קאנטעקסט האט אויך נישט קיין זין.‬

‫ולענינינו, איז דאס אולי א שטיקל רמז אז בעת בריאת העולם, וואס מעשה בראשית איז וואו מען קלערט אויף מציאות הא-ל, טרעפן מיר אז ס׳דא דעם ״אור״ הגנוז מחמת הרשעים...‬
‫והיינו, אז צווישן אן אינפיניט סוּפערטעסק פון א בּיינערי אור וחושך ביז מען קומט אָן צום מושג פון ״2״ קעגנזייטיגע הוכחות, איז דאס א זאך וואס איז שווער משיג ומכריע צו זיין וכו׳ ונגנז הוא מבני אדם. ‬

מי אני האט געשריבן:וז"ל של העיקרים (מאמר ב פ"ו):

אבל לפי האמת אין זה גדר ולא רושם, כי הוא יתברך אין לו גדר, כי הגדר מחובר מסוג והבדל, ומלת 'נמצא' אינה סוג שתאמר על כל נשואיו בהסכמה כמו שמדרך הסוג להיות כן, כי אין בעולם סוג יכללהו עם זולתו. כי לא יאמר 'נמצא' על הא-ל יתברך ועל זולתו בהסכמה, כי המציאות האמתי הוא אליו יתברך, ומציאות שאר הנמצאות קנוי ממציאותו, ואחר שהנמצא אינו סוג יכלול אותו יתברך ואת זולתו, אין לו הבדל.

ועיין ג"כ במו"נ ח"א פנ"ז "הוא נמצא ולא במציאות" ע"ש.

דאס דערמאנט פון דעם פראנצויזישן טעאלאג מישעל דע סערטעא'ס מימרא:
‏

The weight of [God's] transcendence makes any proposition relative, even to the point that the statement "God exists" has to be followed by a denial

והעמיק עוד כי גם המציאות והאחדות לא יאמר בו ית׳ כי אם בשיתוף השם.

ובכלל, באמרינו שהוא ית׳ איננו בלתי יודע, כבר חייבנו שהוא יודע… [ו]מה שיתחייב ביודע, יתחייב ביכול ורוצה ושאר התוארים.

If someone concluded that each of these arguments for God’s existence has only a 13% chance of being true or sound then instead of saying it's still more likely than not that at least one of the arguments is true he should say “It's still more likely than not that God exists.” That the overall conclusion is true. So he's quite right, and Tim McGrew (an excellent philosopher in dealing with probability theory at the University of Western Michigan) has published an article on this in Philosophia Christi where he points out the power of the cumulative case where it could be that in the case of these five arguments that I typically present (or seven in the Sharp debate) each one has, say, only a 13% probability that theism is true given that argument. And yet the cumulative force of the case would easily satisfy the demand of Sharp that this overall conclusion be highly probable. So he's quite right. That is the power of a cumulative case. You don't even need to present arguments that make it 51% probable that God exists. One could make it 23% probable that God exists, another 25%, another 15%. And yet the overall cumulative probability of the arguments would make it highly probable that God exists

מי אני האט געשריבן:און אז מ׳האט דערמאנט די געדאנק פון גאָדעל׳ס טעארעם, וואס צייגט טאקע אויף די געדאנק פון אומעגליכקייט אלעס צו דערגיין בתוך א סיסטעם, שרייבט דער מאטעמאטיקער דר. דזשעימס סטיין אז עס איז יתכן אז עס איז דא א פּרוּף וואס צייגט אויף דאס אומעגליכקייט צו אויפווייזן אדער אפפרעגן מציאת הא-ל (איינמאל מ׳האט דעפינירט די פּראַפּערטיס פונעם מושג ״ג-ט״):

Both sides [theism and atheism] have been so busy trying to construct proofs supporting their case that it seems to me they have overlooked the obvious. Once the attributes of a deity are precisely defined, there may be a proof that it is impossible to prove the existence or nonexistence of such a deity. Alternatively, the deity hypothesis may possibly be shown to be independent of a set of philosophical axioms, adjoining either the deity hypothesis or its negation to those axioms leads to a consistent axiom set

Mathematicians don’t have much use for divergent series these days, but physicists seem to have a higher tolerance for them. String theorists actively like them. Of course they do

Many people find fractions and decimals confusing, counter-intuitive, and even scary. Consider the story of the A&W restaurant chain’s ill-fated third-of-a-pound burger, introduced as a beefier rival of the McDonald’s quarter-pounder. Many customers were unhappy that A&W was charging more for a third of a pound of beef than McDonald’s charged for a quarter of a pound. And why shouldn’t they be unhappy? Three is less than four, so one-third is less than one-fourth, right?

Well, that’s what many of those aggrieved customers told the consultants who had been hired to find out why A&W’s “Third is the Word!” innovation had gone so disastrously awry. But I wonder if those customers were rationalizing (sorry…) after the fact. Maybe some of these people had had such bad experiences when learning about fractions in school (the awkward fraction 1/3 in particular) that they preferred to avoid eating at establishments that triggered their math anxiety

the Archimedean property: Given two counting numbers m and n, no matter how disparate in size they are, if you add enough m’s together you can get a sum at least as big as n, and vice versa. The older I get, the more profound I think the Archimedean Property is, not just as a mathematical assertion but as an assertion about the observable universe. We study quarks and we study galaxies, and they’re very different from each other, but they occupy a common scale, with human beings somewhere in the middle. Maybe there are things that are infinitely smaller than quarks or infinitely larger than galaxies, but how could we ever come to know about them? It seems to me that the Archimedean property of the counting numbers in a way corresponds to fundamental limits on our ability to probe the universe with our finite bodies and minds

I asked ChatGPT, the modern apotheosis of unjustified self-confidence, to prove that .999… is less than 1. Its reply began “Here is a proof that .999… is less than 1.” It then proceeded to show (using familiar arguments) that .999… is equal to 1, before majestically concluding “But our goal was to show that .999… is less than 1. Hence the proof is complete.” This reply, as an example of brazen mathematical non sequitur, can scarcely be improved upon

Curiously, shortly after ChatGPT gave me this answer, the chat session terminated unexpectedly, and when I started a new session and repeated my question, ChatGPT gave me a more sensible answer; no matter how strongly I prompted it, it wouldn’t repeat its earlier bogus answer. I know ChatGPT is just a predictive language model, but it was hard no avoid the sensation that this predictive language model was ashamed of its earlier response

Ramanujan himself didn’t mention the zeta function; he just did some manipulations of the series. But which manipulations are we allowed to do? There’s the rub. There are ways to manipulate Ramanujan’s sum so as to lead to different conclusions than Ramanujan’s. For instance, there’s a way to “prove” (*) that 1+2+3+4+… is 0, and there are infinitely many ways to “prove” that 1+2+3+4+… is −1/8. There are good reasons to prefer Ramanujan’s manipulations to these, but anyone who shows you Ramanujan’s derivation without explaining why his way of juggling symbols is profound and the others are mere curiosities isn’t telling you the whole story

I think it’s misleading to say that 1+2+3+4+… equals −1/12. It would be better to say something more like “the divergent series 1+2+3+4+… is associated with the value −1/12” or “the zeta-regularized value of the series 1+2+3+4+… is −1/12” or “The Ramanujan constant of the series 1+2+3+4+… is −1/12.” Phrasing the result this way isn’t as catchy as asserting equality, but it’s more honest (while at the same time more respectable-sounding than “1+2+3+4+… wants to be −1/12”)

(*) If S = 1+2+3+… then 2S = 1+1+2+2+3+3+… = 1+(1+2)+(2+3)+… = 1+3+5+… = (1+2+3+…) − (2+4+6+…) = S − 2S = −S, and 2S = −S implies S = 0

Here’s another fun linguistic-cultural perspective on diagrams in Greek geometry. The language in which Euclid describes constructions is quite odd. “Let the circle ABC have been described.” The language of Greek mathematics “makes the author and temporality disappear from a proof,” as one historian has put it. Euclid is not saying that he’s drawing the diagram, and he’s not telling the reader to draw the diagram. He’s just sort of commanding the diagram into existence

You know the book of Genesis in the Bible: “Let there be light,” God said, and there was light. Euclid uses literally the same kind of construction. It’s exactly the same verb form as in the Ancient Greek version of the Bible. Just as God makes heaven and earth by merely pronouncing that they exist, so Euclid makes geometrical objects appear just ordering them to be. It’s not “I draw” or “you draw” but “let it have been done”

You could read this as supporting a Platonic conception of mathematics. Euclid is distancing himself from actual drawing. The objects of mathematics just are. They are not something you or I have to make

But here’s a counter argument to this interpretation. Netz argues that actually Euclid’s grammatical construction merely reflects a purely practical circumstance of the Greek tradition. Namely, that Greek mathematicians had to prepare their diagrams in advance due to technical limitations of the visual media available. Here’s what Nets writes:

“Of the media available to the Greeks none had ease of writing and rewriting. [Standard media were papyri and wax tablets, and, for larger audiences, such as Aristotle’s lectures,] the only practical option was wood painted white. None of these [ways of representing figures] is essentially different from a diagram as it appears in a book. The limitations of the media available suggest the preparation of the diagram prior to the communicative act - a consequence of the inability to erase. This, in fact, is the simple explanation for the use of perfect imperatives [such as] ‘let the point A have been taken’. It reflects nothing more than the fact that, by the time one comes to discuss the diagram, it has already been drawn”

That’s Netz’s interpretation, and if he’s right then Euclid’s grammatical choice reflects only incidental cultural circumstances and says nothing about philosophical commitments. “Let it have been done” just means “I did it yesterday”. It doesn’t mean that geometry is set apart from concrete action and that doing has no place in mathematics