# Politics of Mathematics

Proof by contradiction follows from the law of excluded middle ((p or not-p) is universally true), the characteristic law of classical logic. The basic reasoning of proof by contradiction is: in order to prove p, we prove a contradiction from ~p (not-p).

“G. H. Hardy described proof by contradiction as “one of a mathematician’s finest weapons”, saying “It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.”[1] ” Wikipedia on proof by contradiction

Interestingly the United States has become known for using exactly this type of logic ever since the nuclear era began. Of course we have the Cuban missile crisis where Kennedy offered ‘the game’ in this case ‘the destruction of the world’ as a possibility to Khrushchev.

Destruction by climate change has increasingly entered into our calculations, so that during the 2016 elections we were faced with a choice between the republican party (who are blocking the Paris Agreement from really taking off) and Hillary Clinton. And the result seems to be that life in America is so bad that they are willing to try the alternative.

And logic can help, not classical logic, but constructive logic also rejects the Law of Excluded Middle and proof by contradiction. Changing our minds (our reasoning) is a very important ingredient in changing our politics. Constructive logic takes the perspective of building things as more important, more fundamental, than universal or global laws. Because of this, the “world” of a constructive reasoner is only this or that construction, and the progressive stages of continuing construction. In this sense the geography of construction is “incomplete.”

This incompleteness is fundamental and draws the rest of mathematics into reformulation (Bishop 1967). Things like space and time need a new kind of real numbers to describe. If our mathematics were different, the upgrading of nuclear weapons and their resultant destructive power by a factor of 3 would not be considered “an incredible feat of human intelligence” (Noam Chomsky) because we would have a different understanding of intelligence (reasoning power), and empty-headed “intelligent” weapons developers, along with coldly calculating CIA heads would be following different rules. Offering the game would no longer be considered “one of our finest weapons”.

# Synthesis of Vagueness and Logic

by Andrew Nightingale, November 14th, 2559

CC: Dr. Khajornsak Buaraphan, Dr. Parames Laosinchai, Dr. Patchayapon Yasri

The problem with Heideggers “enframing” attempt—that science enframes nature and in any frame there the phenomena are still vague—is that certain kinds of vagaries do not entail paradigm shift. “Though discovering life on the moon would today be destructive of existing paradigms (these tell us things about the moon that seem incompatible with life’s existence there), discovering life in some less well-known part of the galaxy would not.” (Kuhn 1970, p 95) However, some vagueness does warrant paradigm shift, because “Ambiguity [between terms and the world], … turns out to be an essential companion of change.” (Feyerabend 1999, p 39)

Precision, on the other hand, is not an argument in favor of a theory because “In fact, so general and close is the relation between qualitative paradigm and quantitative law that, since Galileo, such laws have often been correctly guessed with the aid of a paradigm years before apparatus could be designed for their experimental determination.” (Kuhn 1970, p 29) So that the measurements are predicted with intense precision, and then the experiment carried out is an elaborate, highly overdetermined one that has only one possible interpretation within the paradigm.

Vagueness is apparent to the naked eye, but it is traditionally opposed to what can be grasped rationally. “In general, Leibniz had followed the other great rationalists in interpreting perception as a confused form of thinking. Like Descartes, he had treated the deliverances of the senses as sometimes clear but never distinct.”(Walsh; Edwards Ed. 1972, p 307) However, vagueness is a clear and distinct concept, and it seems that it also is in complete agreement with the “deliverances of the senses.” Thus, in the sense of mathematics that Whewell and others held, vagueness is a truly mathematical one, that is,

“…in mathematics there was no difference between objective reality and subjective knowledge; the human mind was completely in tune with external fact.” (Richards 1980, p 362) Rational thought and empirical observation are brought together into one concept: vagueness. This old idea of mathematical truth has changed drastically now. With Godel’s theorems, it became clear that an absolutist (that is mathematics is absolutely true and unchanging) view became untenable. One stronghold of the old sense in which mathematics is true (Whewell’s) can be found in the mathematician Brouwer’s intuitionism. According to Brouwer (and Kant before him), the experience of time is accessed to fill the empty formalisms of mathematics, giving it meaning and truth. Vagueness is another source of mathematical truth. It may be that vagueness between two things is present in Brouwer’s intuition of a “twoity,” the beginning of intuitionist arithmetic.

What do I mean by vagueness? The ancient representation of vagueness is the problem of the heap of sand. When you have a heap of sand, you have a relatively safe inference that if you take one grain from a heap, then you will still have a heap. As the story goes, eventually taking grains of sand will show this if, then statement to be faulty because you will no longer have a heap of sand. Why does classical “if, then” fail us here? There is an analogue between the heap of sand example and with the calculation of a real number according to a rule. Also, this question gains importance when reflecting that “Logical consequence [the if, then] is the central concept in logic. The aim of logic is to clarify what follows from what. – Stephen Read, Thinking about Logic [99]” (As quoted in Beall, Restall 2006, Kindle Edition) According to Beall and Restall, logical consequence can be clarified in more than one way, giving rise to more than one equally valid (if applied in different situations) formulation of logical consequence. “We must reconcile ourselves to the fact that every precise definition of [logical consequence] will show arbitrary features to a greater or less degree.” (Tarski as quoted in Beall, Restall 2006)” Additionally probability theory is not a solution to the vagueness of logical consequence, because

…probability theory might provide a canon for evaluating degrees of belief, … Nonetheless, probability theory cannot be a complete answer here, for we also make assertions and denials (and hypotheses and many other things besides), and these may also be evaluated for coherence, using the norms of deductive logic. In particular, we hold that it is a mistake to assert the premises of a valid argument while denying the conclusion… (Beall Restall 2006, Kindle Edition)

The solution to the vagueness of logical consequence, rather, lies in logical pluralism. Logical consequence brings true conditions to their true conclusions, but logical consequence itself is conditioned, and ultimately forms the structure of what can be intelligibly conditioned. Since phenomena are inherently vague, and logical consequence is vague until arbitrarily made precise, there is no clear difference between form and substance, ideas and things.

references

Beall, J. C., & Restall, G. (2006). Logical pluralism. Oxford: Clarendon Press.

Edwards, P., & Walsh, W. H. (1972). The encyclopedia of philosophy (2nd ed., Vol. 4). New York: Macmillan.

Feyerabend, P., & Terpstra, B. (1999). Conquest of abundance: A tale of abstraction versus the richness of being. Chicago: University of Chicago Press.

Kuhn, T. S. (1970). The structure of scientific revolutions. Chicago: University of Chicago Press.

Richards, J. L. (1980). The art and the science of British algebra: A study in the perception of mathematical truth. Historia Mathematica, 7(3), 343-365. doi:10.1016/0315-0860(80)90028-2

# Losing the Language-Game

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.

# Kant and Continuity

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.files.wordpress.com/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.

# Order and the Real Numbers

“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.

# Possibility and Peirce

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.

# The problem of difference

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. -><-

# How Vagueness Reveals and Precision Conceals

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.

# Vagueness in Mathematical Terms

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.

# Possibility and Realism

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)