“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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My daughter was counting hooks in a line, but she did something interesting. As she counted hooks, she counted the spaces between hooks, so that the next space had the same number as the previous hook. Perhaps the ordinary way to think of the space between hooks is the same—as zero—that doesn’t change the count, but she felt, quite unprompted by me, that each space was unique in its place between 1 and 2, 2 and 3, etc—That it had an identity. Normally identity in math is denoted with “=”. Expressing the identity of 1 is done by writing “1=1” or . A space is the opposite of an object, but the difference between hook 1 and hook 2 could be thought to have an identity, because it can be identified with its context between hook 1 and hook 2, for example, and could be written  . It was in Plato’s The Sophist that it was offered that not-being be the spaces or differences between beings. The inequality, or difference, between 1 and 2, , or between 2 and 3 are such spaces.

I am contrasting a definition of number as a difference with a common belief that a number is a class or likeness of members. Normally the number three is essentially a set of three members, where the members of the set could be any three things we wish to collect or associate. However, we run into difficulty when five year old children count spaces instead of objects. What do the spaces have in common, how can they be identified, except as differences between objects? And in that case, how are we to deal with the idea that three “spaces” share a likeness, when all they are is purely a difference?

The point of this is to argue that different differences (or inequalities) are different from each other. Put simply

As argued in a previous post (https://questionsarepower.org/2015/01/31/degrees-of-difference/), the difference between 1 and 2 is different from the difference between 2 and 3, even though 2-1=1 and 3-2=1, we have to keep the context of the first and second subtractions with their results.

To return to that argument briefly, observe how the difference between age 1 and 6 is much greater than the difference between age 40 and 45, even though the distance between each set is 5. Further, one difference between and circle and a square is that the square can be triangulated while the circle cannot, but this difference can be turned into a similarity between a square and a strict rectangle. This would suggest that a difference “=” a likeness. Back to our inequality above, that we could denote the first difference “2-1” with ,and the second difference “3-2” with ,. If we were to lose these contexts and allow both to be reduced to the same “,” contradictions would follow. For example if we were to lose the contexts in the inequality

by writing simply

In English, the idea that something “does not equal ”, reduces to “=”, so we replace two of the “not-equals” with one “equals”:

Now, what is  equal to?

Since we already know , it cannot be that . That would be a contradiction, but the alternative:

Is also a contradiction.

So we cannot lose our context for differences, and must be aware of the difference between differences. The symbols “” and “1”  are misleading because they erase their contexts including if “1” is a difference or identity. This means, apart from the current context-wanting misuse of the inequality sign, that difference, variation, diversity are not universal, they cannot be generalized. Instead difference is always special to its context. Difference is not a neutral term, because it always implies a discrimination between particular persons, places, or things.

Complementary to this vein, and in defense of its pedagogy, an aging Augustus De Morgan wrote an essay in which he tried to generalize the “=”. (Augustus De Morgan, “On Infinity and on the Sign of Equality,” Trans. Cambridge Phil. Soc. 1871, II:145-189.) The problem he faced was:

On generalizing and replacing 0 with x we run into some trouble. De Morgan’s philosophy and historicity of mathematics was such that he felt great mathematical progress was found in “mangled” (Pickering 2006) algebra. (Richards, 1987) The trouble was not that the steps were incorrect, but how to interpret the steps so that the symbols were about something. Understanding a generalized = is one thing, and it may be that the can be generalized as above and then interpreted so that it is still about something, but I’d contend that the nature of inequality is that is is the opposite of generalization. Generalization, like an umbrella or a set, asserts a higher likeness within its domain. To assert a set A, for example, in which each of its members had some difference or other with the other members, would be like the power set of everything. Finding a contradiction or two in this set (of course it would contain Russell’s Paradox) shouldn’t be too hard.

So I’ll do it here: the set A, whose members have some difference from all the other members of the set, must contain itself (A in A), since A, the set, is not like any other member of A. This means that the property that defines the set and that the members of A share, that of being different, is not shared by any member of the set. -><-

Read more here: https://questionsarepower.org/wp-content/uploads/2016/03/many_roads_from_the_axiom_of_completenes-2.pdf

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People often think that vagueness is bad, a kind of darkness that can never be fully dispelled, while distinction is hailed as the clarifying answer to vagueness. Here is how the reverse is also true: Vagueness is the light and distinction a darkness.

The distinction I pick is not random, but an important part of all other kinds of logical distinction—the distinction between the “if, then”: “→” and the “conclusion” symbol: ⊢. ⊢ is ambiguous, however, and can mean other things such as assertion that a proposition is true and not just being named, or to assert in a metalanguage that the following is a theorem in the object language. Used in our sense here, the good property of the “→” is “true” and the good property of the “⊢” is “sound”. The distinction goes back to Aristotle. The main point is that if we do away with this distinction, call these two symbols the same, an interesting insight can be made—that a sound argument:

A
A→B
⊢B

Can be represented without the ⊢ as follows: [A AND (A→B)] is logically equivalent to [A AND B], so that the conclusion [A AND B]→B is merely a deduction of A from [A AND B]. Allowing a vagueness between → and ⊢ reveals what logical deduction is—it is a cut from a larger whole, e.g. logical deduction is the act of drawing a distinction from the larger [A AND B]. With the introduction of the distinction between → and ⊢ this is concealed:

A
A→B
⊢B

cannot be collapsed into [A AND B]→B. As promised, vagueness reveals and distinction conceals, but not just any concealment, here we have a concealment which allows distinction to reveal, since this distinction is at the root of any further logical distinction.

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A paper about vagueness in paraconsistent logic (Weber, Z. 2010) gave a rhetorical example of the interaction between vague terms and real analysis. Imagine walking down the largest mountain in the known world. Olympus Mons on Mars has a very gentle slope or declivity, so that you can start by describing yourself as “high up” but as you walk down you become unsure if you are “high up” or “not high up”. This is graphable on an xy coordinate system with 0 at the top of the mountain and 1 at the foot of the mountain. The interval of “high up” points has a least upper bound (of x values) called “sup” and the interval of “not high up” points has a greatest lower bound called “inf.” Now we will suppose, as this is a presentation of vagueness, that “sup” is also “high up” (which is reasonable since all the points less than it are also high up) likewise “inf” is “not high up”. We encounter a contradiction in all three possibilities: If “sup”=”inf” then “sup” is both “high up” and “not high up” if “sup” > “inf” then by density of the reals “sup” > z > “inf” with z both “high up” and “not high up”, likewise if “inf” > “sup”.

Now we can put the blame on the vague term “high up” which is clearly not very technical, and go on to fantasize a perfectly precise world of mathematics that should not be sullied by vague words, but such a world is more difficult to defend than the fantasy suggests. First of all, mathematical equations with no tie to real world meanings are widely regarded as ambiguous, a term usually distinguished from vagueness. Ambiguous means there is more than one possible meaning or interpretation. Hesse points out that Socrates’ famous straight stick in water, the apparent bend in the stick was meant to represent falsehood. Now the bend can be represented with an equation involving the theory of refraction:

“sin(alpha)/sin(beta) = Mu”

can be interpreted in other ways besides that alpha and beta are angles and Mu is a constant about air and water— “They might, for example, be the angles between the Pole star and Mars and Venus respectively at midnight on certain given dates; why would not this be a confirmation of the formalism we have mistakenly called the wave theory of light?…” (Hesse as cited Structure of Scientific Theories Suppe 1977, p 100-101)

Now suppose I invent a word that means ambiguously “tall” and “not tall.” Similar words are found in language, for example the Thai “Krup” means “yes” but also does not mean yes. It is used because yes is too strong and seen as insulting a person’s intelligence, and “krup” rather is a polite sound indicating the speakers faith that the listener can figure it out for themselves. I could invent a word “snook” that can mean “tall” and “not tall” in different situations. Is the word ambiguous when it is applied to the one of these borderline cases when “tall” is vague? or does the ambiguity of the word capture the vagueness of the phenomena? Is this a vagueness between ambiguity and vagueness? What does that mean for ambiguity in mathematical equations?

The mathematical part of the wave theory of light, if it is to correspond to reality at all, is vague. Even if we were to abandon mathematical application to science, pure mathematics still uses words such as “continuity” “completeness” and “integral” which are vague notions. In fact, a standard text in analysis will show how mathematical definitions fail to perfectly capture the idea of continuity, with fuzziness in functions getting past the definition, and being allowed to be called technically “continuous”. Without this sparse collection of vague words, math texts would be hopelessly meaningless. The vagueness of the words continuity and completeness are in fact very important to being able to learn and understand analysis.

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The following is a stroll through the interconnections between a locus of concepts: vagueness, clarity, possibility, questions, and belief. The result is an offering that vagueness belongs to possibility while clarity is closer to an opposition from possibility, which goes against some thinkers on the topic of possibility, including Descartes.

“I must confess that it makes very little difference whether we say that a stone on the bottom of the ocean, in complete darkness, is brilliant or not—that is to say, that it probably makes no difference, remembering always that that stone may be fished up to-morrow. But that there are gems at the bottom of the sea, flowers in the untraveled desert, etc., are propositions which, like that about a diamond being hard when it is not pressed, concern much more the arrangement of our language than they do the meaning of our ideas.” (C.S. Peirce ed. Houser and Kloesel 1992, p 140)

First, contrast with Kuhn:

“As the problems change, so, often does the standard that distinguishes a real scientific solution from a mere metaphysical speculation, word game, or mathematical play. The normal-scientific tradition that emerges from a scientific revolution is not only incompatible but often actually incommensurable with that which has gone before.”(Kuhn 1962, p. 103)

The “meaning of our ideas” can be shifted fundamentally, so that what was a mere mathematical play (such as in https://questionsarepower.org/2014/08/) suddenly is taken seriously. What is a joke and what is serious changes, the grave becomes light, uplifting wonderment gives way to cold sadness, and with it our world shifts.

Peirce’s quote is interesting because it brings up the relationship between possibility and realism. Diodorus defined the possible, in perhaps the first definition of its kind: “a proposition is possible if and only if it either is true or will be true.” (Mates 1953, p. 6) Under that assumption, the realist question “If a tree falls in the forest where no-one hears it, does it make a sound?” is relevant; but here we must modify the question to be more difficult. Ask: “If a tree falls into the ocean and is washed to the ocean’s bottom, is it burnable?” The two questions are more related than they may seem. In the second question we have to ponder the meaning of possibility. Are things possible even if they neither are nor will be? In the first question we are asking if something is when its being can only be inferred, not experienced. We associate sound with the falling of a tree to the point of inference, that is how we can make the realist claim that such things happen without our experiencing it. But inference depends on our definition of the possible. For Diodorus, beginning with his restriction on the possible (we may, for arguments sake, suppose that the tree at the bottom of the ocean is neither burnable nor will be burnable) took his own definition of inference from it.

“a conditional proposition is true if and only if it neither is nor was possible for the antecedent to be true and the consequent false.” (Mates 1953, p. 6) Such a definition is more strict than the material implication, so strict that it rejects the realism of a tree falling in the forest and making a sound. With this definition, both “If a tree falls in the forest, then it makes a sound” and “If this tree falls into the ocean, then it is burnable” are false, because it is possible for a tree to fall and not make a sound (maybe a small tree fell a small distance on many feet of snow?), and when I say possible I must use Diodorus’s definition, and say that “at some point either now or in the future, a tree will fall in a snowstorm and make no sound.” This means that realism is entangled in a consideration of possible futures. To say that, at the very least, being able to imagine and execute our next step in our stroll is a least part of this entanglement of realism and possibility.

Also there is the problem of genera. For realism does not exactly refer to actual objects, but to the general principle that things occur “out there” without anyone experiencing them. Stoics seemed to think that genera, such as “a (any) tree” were neither true nor false, since “…the generic Man is neither Greek, (for then all men would have been of the species Greek) nor barbarian (for the same reason).” (Mates 1953, p 35) This puts realism outside the consideration of logical truth and falsehood.

Pierce, on the other hand, uses his idea of possibility to assert his idea of truth and realism. “Our idea of anything is our idea of its sensible effects…”p132 Peirce asserts realism, but says that “the meaning of our ideas” is sensible effects. However, possibility enters again:

“Who would have said, a few years ago, that we could ever know of what substances stars are made whos light may have been longer in reaching us than the human race has existed?…And if it[scientific investigation] were to go on for a million, or a billion, or any number of years you please, how is it possible to say that there is any question which might not ultimately be solved?” (C.S. Peirce ed. Houser and Kloesel 1992, p140)

So that “Reals,” as he calls them, guide our inquiries toward them in such a way that any question is answerable, and questions that are not answerable—such as Zeno’s questions or the Liar, are merely the product of language arrangements. They are not “living doubts” that drive a real investigation towards something real.

Circling back again to Kuhn, what was a mere language game can become a serious crisis for someone: “…the new paradigm, or a sufficient hint to permit later articulation, emerges all at once, sometimes in the middle of the night, in the mind of a man deeply immersed in crisis. What the nature of that final stage is must here remain inscrutable and may be permanently so.” (Kuhn 1962, p 90)

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“House-builder, you’re seen!
You will not build a house again.
All your rafters broken,
the ridge pole destroyed,
gone to the Unformed, the mind
has come to the end of craving.”

-Siddhārtha Gautama (the founder of Buddhism), upon his reaching enlightenment (Dhammapada)

It was speculated by Thanissaro Bikkhu that the house meant selfhood or perhaps entity-hood in the commentary of the Dhammapada.

I would propose a model for logic that is a house. Some logical structures are immense. The light that passes through a window would be Truth, the laws that light follows as it interacts with the building would be the laws of logic, the specific form of this particular building would be the logical statements, determining the way truth(light) moves through the logical structure. The trouble is completing the logical elements- what is falsehood? Obviously it is darkness, but the building would have to have no qualities except its form- no colors, no features, just featureless glass mirrors, otherwise truth would fade as it interacts with opaque surfaces- making truth and falsehood mingle. If the walls are perfect mirrors that propagate the light perfectly, a false space would have to be totally cut off from the light. Hypotheticals would be doors, sometimes open, sometimes shut. The only danger of falling into darkness would be entering through a door and closing it, completely cutting yourself off.

The theory that comes to mind is Anaximander’s, who thought the sun was just a hole in the cosmos, where light could enter from outside the Universe. And why is this ideal of logic impossible in the real world? There are no perfect mirrors, matter has color that absorbs light, making it an intermediate between truth and falsehood. When logic from true principles is applied to real things, interacting with matter, the truth will fade into darkness as the logical statements progress, regardless of how perfectly the laws of logic are followed. If the world of logic were to be perfect, the truth could not originate from our world, or else light that is reflected back out the window of our house would fall, logically, onto ambiguous matter. Thus, passing out the window must lead to a world that looked mostly the same as the building of mirrors.

With the modern conception that words can provide totally transparent access to an object, matter would be the only medium between truth and falsehood, but words simply aren’t transparent. They grow out of metaphors, (as argued in the essay linked in my first post) the word “be” grew out of a Proto-Indo European root which also meant grow- so that someone who is aware of the ancestry of words would have resurrected the feeling of metaphor in the word “be”, coloring the word, giving it a connection that is warranted because “be” would not be what it is now without a fathering metaphor: “being is growing”.

And the design or form of this fun-house of mirrors? Would it carry nameable concepts with it, concepts one would come to know or feel by living there? It would if it had any architectural design. How is this different from allowing a word, or a sign for an idea or feeling, into our logic?

The house of logic cannot allow matter, words, or form, except in a part of the house that is totally dark and without doors- they can be allowed into the part sectioned off as unconditionally false. Otherwise we are allowing degrees of truth, qualifications of truth, and a co-mingling of truth and falsehood.

The focus of this blog (expressed in the previous post) has changed to looking for systems of truth that gradually and naturally falsify themselves. What if we allowed matter in our house, and accepted gradations of truth? How could Aristotelian logic be modified so that each “step” in a logical progression reduced the amount of truth it propagated? The goal would initially be a logic that is calculable, so while we could take our lessons on how the logical system would be set up from how light interacts with matter, the resulting system would not be realistic initially. Following the logical system leads you out of the logical system, however, since the logical laws are not perfect propagators of truth. The logic I am formulating here, while not realistic, leads into a real world.

Most of this blog seems to be about what I call “mystical” thinking which is offered as anti-logic or contradictory. It only seems that way. While I make claims like stupid is smart, raw is cooked, warmth is cold and any number of other kinds of nonsense, I don’t actually believe these things. The goal is to understand the failure of ideation, and so this post is a Buddhist disclaimer. Like a human body, ideas are beautiful in some ways but also stink in some places and have disgusting aspects. There is no escape from this problem except to strive away from ideation. I show paradox in the most eternal aspect of our knowledge- mathematics. It is a tremendous egoism to believe in mathematics in spite of these paradoxes. I do not think we should accept the failure of our minds and efforts to grasp the truth, but rather to reject paradox, mathematics and logic in its present state as a way to understand truth.

I would like to begin a new direction for this blog, guided by questions like “What conception of number will lead you naturally to reject the conception of number?” “How to make number imperfect in a gradual, pedagogical way?” Like the dying sound of a gong, good ideas lead you to peace of mind.

Numbers seem to march on towards infinity in a regular and perfect progression, like a straight highway to the horizon. However, the size of the highway changes as the eye looks into the distance, the distant numbers are different from the near ones. It is very much related to the relative difference between large numbers and small numbers- so that the difference between 1 and 2 is relatively larger than the difference between a thousand and 1,001. This aspect of number should not be concealed, but suggested rhetorically at every turn. The idea that numbers progress evenly and regularly is an abstraction that forgets the “size” or “numerousness” of the number. Without this context of the size of each number in the progression, the progression eventually loses cognizance of its subject – number. Eventually what the progression is about can barely be called number at all. The concept of the natural numbers, its progression of “always one more” does not live forever, but degrades and loses its meaning. Number theory is mortal.

Posing that an improbable event is a chaotic one- that when something happens that probably should happen it is at least in a certain sense orderly, and, when something happens that is very improbable, it is an “accident” as Aristotle would put it.

Now, take an event A that repeats itself over and over. If there is any chance that something else could happen besides event A, the probability that event A would repeat itself over and over would get rather small rather quickly, making a large number of repetitions a chaotic event. But a repeatable event is the very definition of order, the fundament of any scientific theory or law. Even probability itself follows rules, if these rules have any exceptions at all (including the rules about exceptions), using them repeatedly over and over again to measure everything is actually a chaotic way to behave- particularly if you happen to believe that the vast majority of things are unknown- making any known laws very likely to have a lot of unknown exceptions.

A person who really believes probability is the sort of person that thinks he knows most everything, or most everything “is known,” and the possibilities can be counted out and calculated.

Education that involves a lot of repetition actually creates instability, since in a long view of the life of an ‘educated’ person, it is very likely he will reject everything he was made to repeat as fact.

Culture that is not in some sense like a wild beast, will not restrain its people.

Control, following a ruler (whether it is a yard stick or otherwise), comes at a price- that (technically before the control is exerted and over a very long view) the ruled will rebel. But how often?

Belief in rules, in the sense of rules carrying on indefinitely—be it the rule for calculating square-root of two or otherwise, is belief in chaos.