For this month I return to my old tricks: speech against entrenched scientific doctrine. It may seem rash, destructive to do this work. One may counter that the indoctrination that is required is equally rash and destructive. I know that the young often have to learn lessons in an unpleasant way. The reason I write urgently about people’s faith in science is because I am basically a skeptic. I believe that holding a belief as absolutely true, without exception, does not create concord. When people are inflexible about ideas there is more fighting among people, within families, communities and internationally, not less. On one hand the people who strongly believe the doctrine cannot abide people who have another mind on an issue, they can’t even talk to them, thinking them beneath reproach. So far a hive mind has not been successfully implemented (and to my mind should not be implemented), no matter how scientific that hive mind might be. On the other hand, the people who can’t bring themselves to agree with a scientific “party line” tend to look down on themselves as well, feeling that it is their fault and they simply don’t understand. Sometimes they seek out alternative beliefs that are just as conceited as scientific beliefs. As a way to counteract solid beliefs, they create other solid beliefs. They believe that not having an equally entrenched belief to combat scientific belief, that is, to merely argue against a scientific belief, is destructive. A basic skeptical idea is that solid belief breeds discord and strife.

One way to see the drawbacks in having overconfidence about one’s knowledge is looking at the concept of “mansplaining.” The basic problem with the concept of mansplaining is it assumes that someone or other “really” knows about something. Often engineers and math buffs love their subject because of a love of the obvious. They want to return again and again to what they think they “really” know, like recounting the gold coins they’ve collected. And the person on the receiving end of this type of personality gets annoyed either because they don’t care to know, or feel that they really know, and the person mansplaining doesn’t “really” know. So we’re in a contest of who knows better. The skeptic completely avoids this contest, because she isn’t sure if anything is”really” known. The only thing skeptics believe, the only thing that keeps skeptics from nihilism, is they acknowledge that there are certain impressions in the present moment; they do not commit to where the impressions come from (external objects, or internal thoughts, or somewhere else), what it is that receives this impression they call the soul, but what the soul is I don’t know. They avoid a philosophical point of view on what these impressions (Greek phantasiai) are. The main goal of the skeptic is Ataraxia, which is peace of mind from not accepting any dogmatic doctrine. By thinking carefully about pro and con of various dogmatic doctrines, effectively counting the gold coins, getting involved in the richness of one doctrine, and then looking at a counter belief, counting the gold coins and richness, the skeptic can’t decide between the two piles of gold, the two doctrines. After doing this type of comparison a lot, the mind in Ataraxia becomes like a fortress, and a mansplainer will have almost no hold on such a skeptic, even if the topic is new. They may observe that the mansplainer appears rather rash in deciding they know so much, but they will not be moved to believe what the mansplainer believes because they have already weighed conflicting beliefs, nor will they be moved to criticize the mansplainer, because the skeptic doesn’t have an alternative belief to defend except what seems or appears to them through the senses/mind in the given moment.

My argument here seems to be an argument against Newton’s laws of physics. That the theory is “false.” However, I am not making the claim that Newton’s laws are false, I am rather asking if Newton’s laws of physics make sense–if Newton’s laws are neither true nor false. Like the person on the receiving end of a mansplainer, I feel merely puzzled, at a loss, unsure if I understand. Unlike people in physics classrooms who in the end look down on themselves deciding “I can’t do physics” or “I can’t do math” I believe anyone taking the position that they can’t make sense of these “laws” is a respectable position to take. This is the reason for the heavy use of the word “Seems.” It may be assumed that “It seems to me” can be added to all of what follows, and everything in this blog.

Newton’s laws of physics are only true in a perfect vacuum. The funny thing is that the basic notion that defines mathematical space, the open set, seems to assert that there is no such thing as a perfect vacuum. So, if Newtonian physics were logical, it would have to either abandon mathematical space, which would do away with line and point the way it is used in physics texts, or it would have to abandon Newton’s laws. The problem I’m running into is that the open set, the basic building block of mathematical space is defined by starting with a point, and then asserting that any “neighborhood” (or circle if you like, but circle depends on point and space already being defined) around that point, no matter how small, contains another point inside the neighborhood (circle). It is clear that space must have points in it, but could a point or a really small neighborhood still be considered empty space? It seems to come down to the question: What is a point?

The common understanding is that a point is a place, not a thing, and that it is such a small place that nothing can be in that place, because if something could be there, say the smallest neutrino, to say the least, there would then exist something of no size. Could there be a place (indeed many, many places) where only nothing can possibly be? It seems to be a contradiction in terms. A place is a place where something could be, and if nothing can be in this place, then it is no place at all. If we did away with the idea of a point to define the open set, say we used progressively smaller circles or neighborhoods instead of points…. that would result is a very different understanding of what is a circle or neighborhood, basically what space is would change: this new idea of space would not be ordered by the real numbers because they are defined in space as points. Instead space would have to be ordered by neighborhoods, if at all. This may be possible, and even worthwhile, but for the present it would make mathematical space stand on its head.

If there is a smallest particle, say take the smallest subatomic particle, or whatever, the smallest neutrino, I don’t know, and then take a neighborhood that is smaller than that, it seems then you have a perfect vacuum…assuming you can still locate points in this very small neighborhood.  That doesn’t tell us what points are, but at least we can then say that there are very small neighborhoods of perfect vacuums where Newton’s laws can hold true. But to get to this conclusion we have sacrificed a lot: we have admitted that we don’t know what a point is. It seems that any neighborhood smaller than the smallest particle would have the same problem a point has: it would be a place that only nothing can possibly be in, which seems to be no place at all. Also, we have assumed we know what the smallest particle is.

It is safe to say that we can never know when we’ve found the smallest particle, because there will always be sizes beyond our reckoning.

Finally, it seems we are lucky to find a truly empty space large enough to claim that Newton’s laws are absolute and inflexible truths (in such spaces). Such a space is a situation that doesn’t matter much. In the world of breath and bodies, Newton’s laws of physics are approximations of more or less pragmatic use, not absolute truths. And mathematical space doesn’t seem to make sense either. Neither mathematical space nore Newton’s laws ought to be believed inflexibly as though other ideas are beneath reproach

Science is magic in the same sense that knowledge is power. To say knowledge is power we at least need to know what knowledge is. Not easy. After that it is pretty much impossible to know what power is. For example, to squeeze the problem down as much as we can, take potential energy. Where is the energy? The simple example is a round rock at the top of a hill. It has the potential to roll down the hill. It has this potential because it has not rolled yet, so already we get the idea that power is not, but there are so many other situations that could also create potential energy, so many that potential energy is really impossible to define. Thats just one small part of what power could be. So to say knowledge is power, or even Foucault’s power is knowledge, is very pessimistic. It assumes that power isn’t anything more than what we know about it. How do we know that? We don’t. We just sort of wish our knowledge were that magnificent. Or in Foucault’s case, we wish our power could always be known, maybe by the magnificence of our power.

So science has the same relationship to magic as knowledge does to power. It is silly to say as soon as it becomes a science it is no longer magic. That is merely an uninsightful, semantic argument. But magic could be so much more than what has been reduced to a science. We choose to keep our eyes where the flashlight beam in the dark is shining, maybe because we like to see (know), maybe because the vague, shifting shapes we would see in the dark, if we looked beyond our beam of light, are too frightening.

I was in conversation with my brother many years ago and he was talking about an experiment where social pressure could make people believe that the longer line was the shorter, and the shorter the longer. I believe his point was that when social reality doesnot correspond to physical reality, the social reality is wrong.

I pointed out that other contradictory ideas, for example, that two things can be both alike and different, so a triangle and a star can be alike in being polygons, but different in how many angles they have, is also a contradiction, much like the longer line being the shorter and the shorter, longer. Admittedly they are slightly different because “like” is a two-way relationship while “larger” only goes one way, from the larger to the shorter, but that there is another contradictory relationship that we find acceptable is enough.

Much later my brother argued that paradoxical relationships of likeness and difference can be resolved by making them a matter of degree. Two things are “like” 30% and “unlike” 70%, for example.

The problem with this is that = and ≠ are related to likeness and difference, they are types of likeness and difference. Equality is a type of likeness, and ≠ is a type of difference, and they are used in analysis books to build the concept of number. We can’t have the natural numbers without the idea that 1+1=2. So = and ≠ are prior to matters of degree. To be clear, if = means strictly identity, so two apples are not equal to each other (if they were, they would be both = and ≠). One apple is equal to itself. Now, evoking Wittgenstein, we would not bother to say, or be interested in the slightest in, 1+1=2, if 1+1 and 2 were not slightly different from each other. The same way we never say “This apple is this apple”. One could say that 1+1 is the act of grouping two apples, and 2 is the apples already grouped. Looked at in this way 1+1 ≠ 2 ; 1+1 and 2 do not have the same identity.

Because the same problem of likeness and difference arises with = and ≠, we cannot use degree to solve the problem and use = and ≠ to solve what we mean by degree, that is circular.

It may be that this is related to Buddha’s idea of anatta (not-self). The parts of one’s self do not add up to the self we perceive, yet those parts, we say, are what the self is. Buddha uses exactly this argument about a cart– that no-where in the parts of the cart do we find the essence of the cart: 1+1 does not contain the idea of 2, or as Western philosophers including Kant have noticed, 1+1=2 is synthetic.

This is a good thing. Remember Buddha’s talk of the “Deathless” is beyond duality, beyond death and life, self and not-self. If the concept of identity were defensible, or a duality such as = and ≠ could create a system of numbers that could fully describe reality, there would be no escape, no ascendance to the Deathless.

Rest, and “the rest”, is an ultimate concept upheld in the dissertation, and it has serious consequences for logic as we know it, and seems to stand in opposition to pragmatism. Rest, however, is not as easy as it sounds. Buddhist monks do quite a lot of work for the sake of rest. As I got older I learned that rest is hard, because you have so many people around you and before you, working and having worked so hard. If you don’t wear the orange robe of a monk, people will look down on you for, not just resting, but striving to uphold rest as a concept that is defensible.

Ultimately we work for the sake of rest. We want to “have done” something so that we can kick up our feet and relax with some entertainment. Of course it is also true that we rest for the sake of work, but “rest” seems to have fallen as a concept and “work” was in the ascendant.
I practice meditation 3-4 hours a day, and I have discovered that to meditate successfully one must also practice ‘sila’ or virtue, such as the 8 precepts in Buddhism. Not lying, taking care not to hurt little critters, helps your concentration inside and outside meditation. As Buddhist monks propose, it must be realized that “striving for peace” can only be truly successful if such striving is also restful. To have rest be a means as well as an end, requires ‘sila,’ at least the 8 precepts, Here is a good explanation of the 8 precepts,

The Eight Precepts:

1. Panatipata veramani sikkhapadam samadiyami

I undertake the precept to refrain from destroying living creatures.

2. Adinnadana veramani sikkhapadam samadiyami

I undertake the precept to refrain from taking that which is not given.

3. Abrahmacariya veramani sikkhapadam samadiyami

I undertake the precept to refrain from sexual activity.

4. Musavada veramani sikkhapadam samadiyami

I undertake the precept to refrain from incorrect speech.

5. Suramerayamajja pamadatthana veramani sikkhapadam samadiyami

I undertake the precept to refrain from intoxicating drinks and drugs which lead to carelessness.

6. Vikalabhojana veramani sikkhapadam samadiyami

I undertake the precept to refrain from eating at the forbidden time (i.e., after noon).

7. Nacca-gita-vadita-visukkadassana mala-gandha-vilepana-dharana-mandana-vibhusanathana veramani sikkhapadam samadiyami

I undertake the precept to refrain from dancing, singing, music, going to see entertainments, wearing garlands, using perfumes, and beautifying the body with cosmetics.

8. Uccasayana-mahasayana veramani sikkhapadam samadiyami

I undertake the precept to refrain from lying on a high or luxurious sleeping place.

Yep, if you really want rest, you have to give up that entertainment you were looking forward to after work. All of the 8 precepts are a matter of ‘abstaining’ from certain actions and speech. There is nothing to do. You have to spend your restful hours striving for true rest.

(footnote:)

“Shapiro uses an example called the “forced march” to display the power of his theory to handle vagueness with “competent speakers.” The forced march is done by a group of such speakers who examine the hair on the heads of 2000 men arrayed from entirely bald and progressing gradually to Jerry Garcia. They must reach a consensus each time, and Shapiro is concerned with the eventuality that, starting with Jerry Garcia, and inferring that a a small tuft of hair less on the next man preserves hairiness, the inference will break down and the competent speakers, one by one, will be filled with conflict until they decide that one of the men is bald (well before they get to the entirely bald man at the other end of this continuation). Shapiro argues that on breaking the inference with a particular man (lets call this man B(a)), classical logic requires that this break spreads to the men near B(a), so they are now also “bald” even though some of them were only a moment before considered “not bald.” Interestingly where exactly this spread of baldness around B(a) ends, and end it must before Jerry Garcia, is unknown, or deferred to further investigation. Needless to say, the deliberation of competent speakers on which men are bald and which men are not is neverending. Instead of admitting that logic is wrong, we are directed to work and deliberate on the problem forever, and if we ever throw up our hands and stop working, the consistency of classical logic would be questionable. (Nightingale 2018, pg 12)”

I believe at this present time we should debate the idea of rest. Our reaction of anxiety about resting, almost automatic, drives the damage to the environment. Much the same as, given an Englishmen and how they drink alcohol, if the Englishman moved to Mormon country in Utah, The Mormons would condemn the Englishman for their behavior. Likewise how often in Thailand, people will just laugh at the matron of a house who is going around yelling at everybody. If this matron went to the USA, she would find the danger of being confined in a mental hospital.

There is so much work being done that takes its toll on the environment, we are prohibited from mentioning the superfluousness of a lot of the work we are doing now. That said, we should multiply by many times the number of social workers. In general, if we worked less, the earth would heal. There would be fewer products, that we defend by saying “we love our toys.” “Work,” defends every person’s right to take up space and inhabit the world. The earth and humanity cannot regenerate its resources fast enough to feed our work

The proof of substance shows that the universe is not entirely empty. Some people will think this goes against what the Buddha taught, but the Buddha chose to be neutral on the issue of whether anything was endowed with substance, or a “self.” And by self he meant identity. As in his example, even a cart, when examining its parts, the substance of the cart cannot be found, or no part of the cart is the cart itself, and since the cart is made of parts, none of which is the cart itself, the cart has no substance, it is only an amalgamation of parts. So the argument goes.

Now my proof of substance does not go against what the Buddha taught, because it is merely a proof that substance exists. And this proof doesn’t matter much, because a Buddhist should strive to notice emptiness, especially in oneself. This striving can be successful, and we will see how substance is fundamentally connected with ignorance. And how the existence of substance is merely the existence of ignorance, and ultimately results in the first noble truth: the existence of suffering.

1 The first step of the proof is asking: “Is this a question?”

It is like asking if ignorance exists. We cannot decide if it is or isn’t, because deciding would resolve its ignorance, but resolve it in the opposite way we decided: if we decide that it is a question, whether it is a question is no longer in question. After all, how can this question be, if its being were not in question?

2  The next step of the proof is asserting that “is this a question?” while it is undecidable on whether it is a question, is still a tangible thought.

What kind of thought we have when we ask “Is this a question?” is not known. This leads to the third step of the proof:

3 Because “Is this a question?” cannot fit in any form, and it exists, the only option for it is to be substance.

“Is this a question?” cannot necessarily be a question. If it isn’t then what is it? It seems that there isn’t any other kind of “what” it could be, and so it lies outside a what question, even though we know that it is. This is the fundamental ignorance.  As you can see, it is fruitless to argue over whether it exists or not, which is what the Buddha said of the doctrine of self (by which he meant substance). Asserting “is this a question?” is a question yields a paradox, so it may be best to let this question remain in question. In either case, it may be going too far to assert that it is something. After all, a thing is a “what” and this ignorance cannot be categorized or partitioned. Yet it persists.

The goal for today is to prove that magical thinking is rampant in mathematics. First of all lets define magical thinking. I would say that magical thinking is a kind of metaphorical thinking, as in the metaphor “My heart is the sun” only with the added idea that writing these words/making the metaphor exerts towards making the metaphor true to some degree or in some sense. Magical thinking is the claim that saying “My heart is the sun” actually warms my heart.

Now the way that mathematics uses magical thinking is to start with a metaphorical idea of difference. For example, the difference between a “raven” (1) and a “writing desk” (2) metaphorically (not actually) is the difference between the “north star” (3) and the “form of thinking called questioning” (4). It is fairly intuitive that the difference between (1) and (2) is different from the difference between (3) and (4), but mathematics amalgamates all differences together into one concept with metaphor. And it is a particular kind of metaphor that asserts that difference actually works that way.

Even though 3 and 5 are less different (2) than 3 and 9, (6), these differences are not taken into account in the traditional mathematical symbol for difference, the .  Traditionally 3 5 just as much as 3 9, so the identity of difference, , is enforced.

Mathematics asserts an ultimate concept “Difference” that is universal—it works for any situation where there is difference, making any difference “complete” and it does so by metaphorically joining disparate differences. Hence, it falls under my definition of magical thinking.

I am doing the opposite of what Derrida did with his Différance. Derrida added senses to difference, or conflated, allowing it a history and to belong to language, I am suggesting that we subtract, or better divide utterly Difference into differences.

The rest of the sciences follow suit, of course, since mathematics is the language of the sciences. My advisor for my M.S. in mathematics once said “Mathematics is the Poetry of the Sciences.” I would add that Mathematics tends to obfuscate the surprisingly obvious; it is a process that converts potential knowledge into actual knowledge. Poetry deals with knowledge that is naturally obscure, so that beautiful language can be mysterious, profound, even terrifying.

Wittgenstein argues that mathematics is a language game, that is not only based on language but on “forms of life.” And forms of life are rules of the game that are, perhaps socially constructed, but definitely without doubt. They are given somehow and from who knows where. Proving a theorem is to invent a rule that is also without doubt,

“What is unshakably certain about what is proved? To accept a proposition as unshakably certain—I want to say—means to use it as a grammatical rule: this removes uncertainty from it.” (Wittgenstein 1978, 170)

Proving adds to or informs our forms of life. This means that forms of life are created… can they be destroyed? I’m saying perhaps we built the wrong forms of life, perhaps there is still a nagging wonder about the things supposedly settled. And the question begins the process of changing our forms of life. In my first post I proposed bringing in the “?” into mathematical language. As it was laid out in that post, https://questionsarepower.org/2014/08/ the “?” operation would introduce a controlled retreat from mathematical logic. I showed how the “?” can free us from the Liar Paradox, a language game that is an endless cycle in search of a truth value.

The axiom of completeness is exactly such a form of life that should be questioned. It metaphorically closes off the possibility of leaping from moment to discrete moment. The axiom of completeness is like a poetic spell on the mind that prevents natural movement through time, since moving through time continuously, with out the possibility of a leap, generally means “downhill.”  Real analysis is a losing language-game.

I have argued against the “least upper bound;” Dedekind cuts are hardly different from this notion, but it shows the cunning of mathematicians to push their agenda: “Everything is number” by using essentially the same argument in so many different ways. Dedekind cuts are sets of rational numbers with no maximum, that is, if r is in the “cut” called A, then there exists a rational “s” such that r<s, s in A, is similar to saying the number generated by adding another digit to the decimal expansion of r is also in A.

Cuts essentially assert a least upper bound (or the “…” in a decimal expansion) with judicious use of the “<” symbol; it is merely a rewording. Dedekind offered these “cuts” as real numbers and asserted that they exist.

In one of many analysis texts I’ve read,

“The real numbers were defined simply as an extension of the rational numbers in which bounded sets have least upper bounds, but no attempt was made to demonstrate that such an extension is actually possible. Now, the time has finally come. By explicitly building the real numbers from the rational ones, we will be able to demonstrate that the Axiom of Completeness does not need to be an axiom at all; it is a theorem! There is something ironic about having the final section of this book be a construction of the number system that has been the underlying subject of every preceding page…We all grow up believing in the existence of real numbers, but it is only through a study of classical analysis that we become aware of their elusive and enigmatic nature. It is because completeness matters so much…that we should now feel obliged—compelled really—to go back to the beginning…” (Abbot 2001, p 244, Emphasis mine)

Who is being compelled? The book is built so that you have to assume the Axiom of Completeness for a very long and arduous time before they get to the meat of the problem. More importantly, this quote shows the circularity of mathematics, from axiom to theorems back to axioms. Theorems are merely explicit parts of the axioms. With a shuffling of words the Axiom of Completeness turns into a theorem, but certainly the theorem is more explicit and involves more description of what the Axiom is, which is left for the very end of a book devoted to assuming the Axiom.

And how explicit is it really? The cuts are defined as any set of rational numbers with no maximum. Is that explicit? How many sets are like that? And there is an interesting example of a cut: take the set of rational numbers “r” such that  when r is positive, else r is in the set (call this set A). Compare with a similar set where (B). This is exactly the sort of thing that mathematicians enjoy, the “almost false.” Dedekind cuts fail to distinguish these two sets, but Dr. Abbot continues with his claim that Dedekind cuts make real numbers explicit. Strangely enough, while here the difference between < and might not matter, elsewhere in the theorem of completeness it matters greatly. First of all many of the cuts can be distinguished by using < as in the set of any rational r < 2 , which is a cut, but r2 is not a cut since 2 is its maximum and 2 is in the cut. How to describe the cut where square-root of 2 is the least upper bound without this ambiguity? We can’t. Both the symbol square-root of two and  A,  involve algebraic operations without a clear (or even necessarily a single) solution (or lub). We can prove that square-root of 2 is not a rational number, but to say that, whatever square root of 2 is, it is a least upper bound of A is to forget that that is what we are trying to prove.

Now, lets look at the order of Dedekind cuts. For cuts A and B,

AB is defined to mean AB.

Let A be the set defined by , is A a strict subset of B, the set defined by ? It certainly seems like it ought to be. It can be reasoned that A contains less (of what?) than B. We want our different algebraic expressions to have distinguishable numerical values, that is ultimately the motivation for the  real numbers, but in this case we don’t have that. The choice of in the definition is of course very careful. If the definition used < we would have a potential counter example. Figuring out the exact difference in certain cases between “=” and “<” is swept under the rug. Luckily mathematicians can add definitions to counter this particular example, but how many other examples of vagueness are there? The definition of a Dedekind cut is so general (not explicit) that there may be many other problems.

Since we don’t know that B is a cut, we cannot claim that it is a real number nor that it represents a least upper bound. How do we know that A is a cut? We know that either way square-root of 2 is not a rational number. We know that there is no least upper bound “next” to B, that if a number is adjacent to square-root of 2, they are so close as to be the same number. How, then, can square-root of 2 be explicit, how do we know that there is a unique and determinate answer to square-root of 2? The truth is, vagueness sets in as r in B get closer to B’s upper bounds. The upper borderline is vague, like any empirical borderline, but somehow we are compelled into believing there is a unique and determinate upper bound to B.

“Wittgenstein argues that logical necessity—be it computing an algorithm, proving a theorem, drawing a deductive inference, or whatever—concerns the following of a rule. Rule following raises the issue of the compulsion to reach a conclusion that is fixed and, if not predetermined, then at least unique and determinate” (Ernest 1998, p 80) Assuming there is a unique and determinate square-root of 2 makes the expression of such a view merely a language-game, not anything profound. The only way to have a philosophical thought about the language game of arithmetic is to resist this compulsion, and with that act, the entire edifice of real numbers crumbles.

Dedekind cuts are based on a rule for determining a very large infinity of sets; so is the square-root of 2 a large calculation. Wittgenstein wrote about rule following, saying that “What you are saying, then, comes to this: a new insight—intuition—is needed at every step to carry out the order..a new decision was needed at every stage” (Wittgenstein 1953, 75). The compulsion for a student to follow a rule such as calculating the value of square-root of 2 is never fully understood, because of the decision making process, we think we understand the rule and give up calculating, but if we were driven to continue calculating square-root of 2, doubts would inevitably pop up, perhaps merely because we have a life to live besides carrying out this rule, but the objections that pop up will be well informed ones, by someone who has done a lot of exploration into the rule. In Wittgenstein:

“…following a rule and agreeing (perhaps implicitly) to its conventional underpinnings…also involves a decision that the new application can be legitimately be subsumed under existing rules, for rules underdetermine their applications.”(Ernest 1998, p88)

We have invented a substance ‘the real numbers’ that is so well ordered that it can obey algebraic rules without any decision. Thus the decision to adopt the ‘real numbers’ is a very important and determining one. Feyerabend argued that a proof resembles a tragedy. It is internally consistent and inevitable. Comedy, on the other hand, is the stuff of continuity. And the place comedy had in the Greek world was:

Everything commonly realistic, everything pertaining to everyday life, must not be treated on any level except the comic…As a result the boundaries of realism are narrow. And if we take the word realism a little more strictly, we are forced to conclude that there could be no serious literary treatment of everyday occupations and social classes…of everyday scenes and places…everyday customs and institutions…of people and its life. (Auerbach 1946, p 31)

Reducing comedy to tragedy has dire consequences for the reducing culture.

Dedekind played on words with “Dedekind cuts,”  Now, what does he mean that a “cut” exists? Does he mean the knife that cut, or the space between the two parts that were once one? Or does he mean the ground on which the surgery took place, the “operating table,” as Foucault put it?

If he means the space created by a cut, I have already mentioned the trouble with different “cuts”—that different cuts are different from each other, that there exist at least two different real numbers. This means that , and undermines the notion of identity in mathematics.

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In Kant’s Critique of Pure Reason he relies heavily on the continuity of time. For Kant, arithmetic is the a priori form of time, and so it is of utmost importance that arithmetic can describe continuity, or else it is not known whether time is continuous. The function of the Axiom of Completeness is to ensure that all positions in a continuous progression can be expressed with an arithmetical number, thus arithmetic can describe mathematical continuity. What follows is the various ways that Kant depends on mathematical continuity.

1. Kant’s famous question “How are synthetic a priori judgments possible?” (first edition xvii) is given credence because of (to him) examples of such judgments:  geometry and arithmetic.
2. Continuity of time. Arithmetic, to Kant, is the a priori form we impose on phenomena in time. If Arithmetic cannot describe continuity, time cannot be continuous.
3. Application of his categories to phenomena. How can a category, for example, existence, or an a priori form for time, be applied to experience? They can, according to Kant, because of “a “transcendental schema,” which is a “transcendental determination of time.” (Second Edition 181) The schema is “properly, only the phenomenon, or sensible concept, of an object in agreement with its category” Thus being phenomenal means being in time, Kant’s worlds are phenomenal, and the categories find their sense because of primarily our a priori sense of time.
4. Induction over time. Kant with Hume sees that connecting experiences-in-time together (synthesis) is required for empirical knowledge. They both notice that this principle “induction” is not an analytic nor empirical principle. Kant argues against Hume’s belief that experiences-in-time are discrete, instead saying that without continuous (or connected) experience of time we would not be able to synthesize events.
5. Causation, which is just a particular kind of connection between events (a connection according to rules, rules determined by our categories, categories given sense because of the a priori form for time) is possible because events are, in general, connected and continuous.
6. Synthesis and Knowledge. “Synthesis in general is the mere result of the power of imagination, a blind but indispensable function in the soul; without which we would have no knowledge whatsoever, but of which we are scarcely ever conscious.”  (Second Edition 103) When asking how synthesis becomes knowledge, it is in the intuition of data according to the a priori forms of space and time. To Kant, knowledge is not possible without being connected by a unitary consciousness, which in turn requires that time is not discrete.
   

   “For this unity of consciousness would be impossible if the mind in knowledge of the manifold could not become conscious of the identity of function whereby it synthetically combines it into one knowledge. The original and necessary consciousness of the identity of the self is thus at the same time a consciousness of an equally necessary unity of the synthesis of all appearances according to concepts, that is, according to rules, which not only make them necessarily reproducible but also in so doing determine an object for their intuition…”(First Edition 107-108)

7. The self, or unity of consciousness depends on a background of external things at rest or unchanging. Having a “permanence” that requires the a priori form of time for resting to continue.
8. Conservation (“Permanence”) of substance—no total generation nor destruction. Kant again uses the unity or continuity of time, saying that objects at rest, not changing but passing through time continuously, provide the background for grasping change.
9. A realist world. The background provided by this permanence is the basic way we perceive the external world.
10. Because the form of time is an a priori, we have the result that the external world and the ideal world are interdependent.
11. Descartes seems to believe that mind is the first thing we are away of, while it is necessary to infer the existence of external things. Thus external things are open to doubt (indistinct?). Kant argued that because we need permanence of external things to infer a self, external things are not open to doubt. In the “Anticipation of perception” section Kant asks how sensations can have a determinate degree (not indistinct).

Geometry and logic were also generally considered perfectly settled fields in Kant’s time, and were important to Kant’s philosophy. This has changed drastically, however, with the emergence of nonEuclidean geometries and other logics.

To see my argument against the Axiom of Completeness, and against a continuous passage of time, see https://questionsarepower.org/wp-content/uploads/2016/03/many_roads_from_the_axiom_of_completenes-2.pdf

If we hope to retain some of the meaning of mathematics by asserting that arithmetic is about the a priori synthetic intuition of time, we cannot do so by asserting mathematical continuity of time, because mathematical continuity is a spacial notion. For example the Intermediate Value Theorem is meaningless without spacial imagination. How can I be sure that there is an infinitely small period of time, the now, where the past meets the future intermediately?  Such a belief is totally against the experience of moments in time, which are atomic before being divided after-the-fact. Trying to interpret “intermediate value” by referring to non-discrete “real” numbers has already departed from the immediate intuition of time. The idea of completeness of arithmetical numbers, that there exists a least upper bound, does not reveal itself in time, and I believe Brouwer agrees on this point.

I think that the feeling of “flow” from, say, a breeze or putting your foot in a stream is the experience of temporal continuity, but separating its temporal aspect from its spacial aspect is not so easily done, and maintaining ones focus on the experience of purely temporal continuity is quite difficult. Usually one’s concentration breaks and with it, the moment as well. No matter the ability to concentrate, one cannot remain awake forever. Unifying atomic moments after the fact is a feat of the intellect, but it is not the primal intuition framing an event. I am saying that an atomic moment is unbroken and continuous, but a purely temporal intuition of continuity is not captured by mathematical continuity.

Brouwer’s constructive reconstitution of the continuum from the memory of life-moments that have “fallen apart” seems to acknowledge that the intuition of continuity cannot be completely defined. I also feel a deep scruple in pretending that continuity can be defined; such a definition would mean the “end” of time. It is a marvel that mathematicians, who should be less pretentious than philosophers, make such a pretense. The feeling of continuous time should be left to mysticism. For philosophy, atomism of moments is the best we can expect.

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“Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn’t make them right.” (Thurston ed. Hersh 2006, p 48-49)

I remember being greatly troubled, to the point of deep crisis, over how the real numbers were ordered. Can one, for example, say that any two unequal real numbers have a relationship “>” so that we can confidently place them on one or the other side of the “>”? We could if we know what the real numbers are that we are talking about, but there are many real numbers that are undetermined by any easily definable rule. So if I take one of my numbers to be the square-root of 2, and take another number to be just like square-root of 2 except adding one to one of the digits of the decimal expansion of square root of two, the digit being decided randomly from the infinite decimal expansion. Now lets shuffle these two numbers so we don’t know which one is which and compare the two numbers:

Number 1: 1.4142135…

Number 2: 1.4142135…

Do you like my trick? most inspections will yield that the numbers are the same, yet by our rules we know them to be different, do we know one to be greater than the other? Before we answer yes, lets analyze the question: Do we know that Number 1 is greater than Number 2? Do we know that Number 2 is greater than Number 1? So we think an ordering is there, but we can’t apply the ordering to the specific numbers without an arbitrary amount of time to inspect them first.

Another concept of order received much more attention and controversy. The well-ordering theorem asks if we can have a certain kind of ordering, called well-ordering (when a set has a least element) can be created for any set. There were some other details but the point is it was very controversial and eventually proven that no such order existed for the real numbers by Julius Konig in 1904. I am not sure if this was the fiction Thurston is talking about, since according to Mann (https://math.berkeley.edu/~kpmann/Well-ordering.pdf, p2), Konig’s proof was flawed, and the well-ordering principle was proven to be unprovable with the commonly accepted axioms of set theory.

Now, is the fiction that the reals are ordered or that they are not ordered? It seems we don’t know either way, but when I approached my advisor in my M.S. in mathematics program he told me “The Axiom of Completeness orders the Real numbers.” I can see what he meant: the axiom asserts a well ordering of a kind of subset of real numbers: bounded and monotone subsets. There was, however, no hint from my professor that there was any “polite fiction,” and my crisis continued until I rejected the Axiom, and, many years later, found the quote from Thurston today.

The crisis I was having before was not a problem of understanding, but a problem with accepting mathematical theorems as a belief. Peirce argues that the goal of thought, and firstly mathematical thought, is belief, but if belief comes at the cost of understanding, I would rather have understanding.

I would propose that the “…” is not an indication that we know the “rest” of a real number, nor its position in an order, but rather the “…” is an assertion of vagueness about the “rest” of the number and a better symbol to use would be “?” rather than “…”.

The axiom of completeness asserts a kind of empty knowledge of this vagueness. In a sense it “covers” our ignorance with a fact that does nothing for our knowledge, going along for the moment with the Kantian view that arithmetic is the a priori synthetic knowledge of pure time, we have a fact—the Axiom of Completeness—that creates ignorance of our understanding of time.

“Nothing in education is so astonishing as the amount of ignorance it accumulates in the form of inert facts…Before this historical chasm, a mind like that of Adams felt itself helpless; he turned from the Virgin to the Dynamo as though he were a Branly coherer.” —Henry Adams, The Virgin and the Dynamo, 1918

It is my thesis that the tremendous ignorance about time that the Axiom of Completeness creates, metaphorically speaking, is harnessed as fuel for the Dynamo. For an elucidation of the problem of time see “Time, Realism, News” by Kevin G. Barnhurst and me, Andrew Nightingale, in press with Journalism: Theory, Practice and Criticism.

(http://pure.au.dk/portal/en/activities/journalism-theory-practice-and-criticism-special-issue-on-the-shifting-temporalities-of-journalism(63d54d06-0e1a-4512-a237-ae200aafb843).html)

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To Peirce, the clarity of ideas is also important. And clarity is important to possibility. According to Descartes: “To know whether a given idea—for instance, the idea of a circular rectangle—does represent a possibility, one must be able to form a clear and distinct idea of it; if one is able to form such an idea, one has the assurance of God’s goodness that it does represent a real possibility…”(p421 Aune, ed. Edwards 1972)

“…he[Descartes] was further led to say that clearness of ideas is not sufficient, but that they need also to be distinct, i.e., to have nothing unclear about them. What he probably meant by this…was, that they must sustain the test of dialectical examination; that they must not only seem clear at the outset, but that discussion must never be able to bring to light points of obscurity connected with them.(C.S. Peirce ed. Houser and Kloesel 1992, p 125) And the dialectical, involving a deep questioning of a proposition, becomes very important in forming a truly clear idea.

A question “stimulates the mind to an activity which may be slight or energetic, calm or turbulent.” (C.S. Peirce ed. Houser and Kloesel 1992, p128) And the goal of such action involves “a method of reaching a clearness of thought of a far higher grade than the “distinctness” of the logicians.” (C.S. Peirce ed. Houser and Kloesel 1992, p 127)

It seems, however, that clarity should not be associated with possibility so much as with actuality and belief, and vagueness more closely with the question. But admitting that vagueness is a possibility is to admit a monster into our world. And a dangerous monster it is, because many vagaries are found at the boundaries of every idea’s meaning. If possibility is taken in Diodorus’s sense, vagueness must be real now or in the future. And it seems this is true, since it never seems to go away. Our theories fall short of perfect description, and only partially match observation. And, importantly, observations are vague even without words. For example looking at the world with the naked eye is vague when compared to, say, chemical “reality,” and chemical reality suffers vagueness when compared to quantum observations such as the colors of candlelight. In fact, without vagueness nothing is possible, because the motion across a border from the actual to a possibility, looked at closely, reveals vagueness. There in fact is no limit to the progression, no definable borderline, only a painterly wash of vagueness. If all empirical data suffers from vagueness, how can our imaginations be expected to be clear? (Imagination taken in the sense of Bacchelard’s Air and Dreams where the imagination “deforms” empirical data) Take for example the imagination of a mathematical point. If anything is brought to the mind at all, a space or a dot, we have already departed from a precise imagination of a mathematical point, which is a totally determined and perfectly precise position, neither a dot nor an inhabitable space.

Note that my appeal is not to space but to time: the progression from the actual to a possible future. If we hope to retain some of the meaning of mathematics by asserting that arithmetic is about the a priori synthetic intuition of time, it cannot do so by asserting continuity of time, because continuity is a spacial notion. For example the Intermediate Value Theorem is meaningless without spacial imagination. How can I be sure that at there is an infinitely small period of time, the now, where the past meets the future intermediatly? Such a belief is totally against the experience of moments in time, which are atomic before being divided after-the-fact. The idea of completeness of arithmetical numbers, that there exists a least upper bound, does not reveal itself in time. I cannot be sure of such a short period of time, and normally such sequences are only alluded to with graphs in space. I can, however, find, as I strain my intuitions towards ever-smaller moments of time, that eventually the observations become imprecise. When I strain my mind towards possible futures, I find vagueness in their details, as with my failure of a memory. Vagueness is real and finds its analogy in space as well. The limit or least upper bound does is not as certain when grounded in geometry, and becomes a meaningless formalism; a formalism that falls prey to Godel’s theorems.

It has been proposed that wrestling with the monster of vagueness is worthwhile: The formal representation of vagueness—that of a chain of hypotheticals that eventually breaks down because there is no clear cut of point for class membership to end (there is no “least upper bound”) is exactly where mathematics and observation are in total agreement. I am not advocating probability theory as the way to fill mathematics with content, because there is no necessary reason to believe that probability is the answer to vagueness, especially since there are other logics available since logical consequence itself is vague (Beall, Restall 2006).

Peirce seems to ignore the Skeptics belief in Ataraxia, a peaceful state arrived at by abstaining from rashness in belief as a result of inquiry. Personally I find comfort in the idea that we will never answer all the questions, and that there are some questions, while they may seem answered for a while, continually resurface no matter how long are our investigations. It amounts to a kind of faith that the uncertainty of life will go on and essentially be the same as it always has been. Knowing, like Euclid’s parallel postulate, makes us think we are going straight for a while, but as our path gets longer, we begin to observe curves in our path. People enter crisis and paradoxes arise, non-euclidean geometry is born after thousands of years of the problem of Euclid’s postulate resurfacing. The knowledge paradigm shifts, those in Ataraxia are not perturbed.

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