“Only disobedience can threaten authoritarians”</span></div>

https://www.counterpunch.org/2018/10/25/vital-ignored-truths-in-milgrams-obedience-to-authority-studies/</div&gt;

Personally I think culture defends people against authority too. Trump owes some of his success to appearing to defend American culture. Hence MAGA and white male supremacy, but that doesn’t make culture a bad thing to be used against authority. Just because Trump misuses white culture to divide Americans doesn’t mean white culture, and culture in general is a bad thing. What I see in Thailand is a strong sense of culture that makes people agree on issues and behave in ways that unite and protect, regardless of what the “authorities” say. In a sense that amounts to disobedience, but Thai people barely feel that they are being disobedient when they disobey a political leader or policeman, they are just being obedient to their culture. What to do in America with so many cultures and religions? In a sense the divisions in America are very real, not merely caricatures created by a fascist leader.

Feyerabend is my favorite philosopher of science. He says that he wrote his book arguing that science uses no method because he was faced with teaching philosophy of science to people of distant cultures such as Native Americans. He recognized that western science is a product of Western culture. Maybe “science as culture,” or better “science as practice,” is the way to unite the USA. The way to do that is for people who support science to stop pretending its universal, stop making it “above” any discourse with other belief systems. Science has to make peace with religion and culture through dialogue. This means that the ultimate concepts of Difference and Work found in science need to be softened (not to mention the Continuum, which, as Heidegger noticed, replaced all models for struggle, political between the right and left, or otherwise). Being different, or an individual, or working really hard to “make a difference” are not as great as they are touted to be. Dialogue about difference, love based on difference (like between male and female), and faith in difference are easy to allow into this “science as culture.” But also things like friendship and culture, things that really unite people have to be allowed into science as well. “Science as practice” may be the dominant way science unites the USA, and that means using social media, “phones” if thats what they are anymore, and voting machines (not to mention the legal apparatus that elections follow, which owes its authority to its machine-like appearance). Unfortunately mathematical algorithms that are supposed to make the flow of social capital “fair” is too easily manipulated.

In my dissertation, instead of saying “farewell to reason” as Feyerabend did, I try to soften the main ways that science tries to make itself above social, political, cultural, religious discourse. The main ways science tries to do this is through a claim to being logical or reasonable. The other authority claim science makes: that of being empirically based, is not divisive, except that for some reason the empirical basis pretends to exclude accounts of people (except other scientists) and appear to be based on… what? Chomsky reports that science has no clear agreement on what the “Physical” is anymore, so “Nature”? “Difference”? “Work”?. However, classical logic has as a foundational axiom the LEM: that differences between A and not-A, for any A, are absolute and universal. Calling probability a theory and not a logic allows mathematical logic to take its seat as the One True Logic, with its universal and absolute law of Difference, or so it is made to appear in mathematics curriculum that I have seen as a mathematics teacher. My dissertation is meant to argue that the LEM is not acceptable, with the important result that without the LEM, science is able to take on the important role of uniting the USA in “science as culture”.

Science as culture involves adopting other normative values. It would no longer make the pretense of being a neutral descriptor of… again what? not the Physical… Ethical considerations, such as how much work is too much, how much interrogation of the earth is probing too deep, or is too dangerous to the environment, or to our humanity. The atomic bomb, atomic waste, and excessive carbon pollution, and plastic and other pollution, pandemics, and other side effects of a scientific community should render their causes  unscientific.

I am not sure where this outpouring leads. I feel sympathetic to the plight of Americans in the US. Make no mistake: our actions, our beliefs, our inclusions and exclusions, reverberate across the world. But our institutions are more powerful than our individuality. They must be disobeyed. We should be ruled by people, not law, be it scientific law or otherwise.

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.

One project that has been knocking around in my mind for many years (12+) is a news filter that is publicly filtered. By publicly filtered I mean that people are in charge of how a search yields news results, not a mathematical algorithm, not elite editors. I am sure this idea has been thought of before and the reason I think it isn’t already out there is there are powerful people in America who don’t want people to be able to consciously filter their news for themselves. So, I will probably need to hire a lawyer at some stage during this project.

The main innovation is that the program will have more than just a simple “like” way to vote on news. The facebook system is still one-dimensional even though you can choose “laugh” “cry”, etc. For this news filter you can react to a news article more than once using the following dimensions (at first)

Scales (instead of binaries) on the following: “about people vs about institutions” “Democratic vs Republican vs(?Anarchism?)etc” “asks a lot of questions of the reader or tells them statements” “How factual vs how interpretive” (and a separate discussions of the facts and the interpretations available for comment) Also just a rating of how much they liked/disliked the news article. Perhaps you could add the conventional classifications “world, local, science, etc” In addition, I’d like to have people offer and vote on other classifications. Also they can vote on news outlets and filter according to that as well, and the votes could be revealed publicly, so people actually know where most others are getting their news, or like getting news.

There will be two settings: an entry setting that allows you to vote on any news article online, and the other setting is the search setting, that allows people to search based on these votes putting in the search criteria (the same criteria that is voted upon in the entry setting).

Here is a very preliminary sketch. Depending on the amount of donations I can make this much better or hire to work on it.

This is where you come in. I am putting a donation button below, donations will let me know how much time I should devote to this project, and how easily I can argue with my wife that time should be devoted to it. I promise to do this project, and that it will remain 100% free to use. However how pretty, how streamlined, and how much time it takes to finish this project depends on the donations I receive. Thank you

click the link and type in zweihanderdawg at gmail dot com to donate to my paypal. Thank you.

cashtag:
BarnhurstNews

and the most expensive, in that it takes the largest bite out of your donation, is a gofundme:

https://www.gofundme.com/barnhurst-news

Try the Logic Puzzle Program

So, I feel the need to defend the uses of of “=” and ““. Mainly because I am differentiating uses of but not differentiating uses of “=”. If I differentiated, say the identity “A=A” from the identity “B=B” it would mean not that these were different identities, but that there were two kinds of identity. That is a can of worms I am not ready to open yet. But the choice makes more sense if you think of the concept of equals, including identity, as being the opposite of making distinctions, which is what identifying kinds of identity is. This is what makes equals universal, while not-equals is always special, even though I am aware that this undermines the concept of difference, making (allowable) inconsistencies in my previous post.

We can say, for example that 1+2=3 and 2+3 = 5 and then that 1+2 2+3 which in effect means that the two equals signs are for different equalities. No problem there, and the difference is kept with the symbol , which is the way it should be. But how can I defend that there shouldn’t be different kinds of identity, where there should be different kinds of distinctions? Well, is the way that “A=A” any different from the way “B=B”? I would say no. And, again appealing to the intuition: There is a big difference between the way a question is different from a neutron star and the way an lemon is different from a lime. So I am committed to the idea that there are different kinds of differences, but there is only one kind of sameness.

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 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.”

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”.

by Andrew Nightingale, November 14^th, 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.

Read more: https://questionsarepower.org/2016/08/19/the-problem-of-difference/

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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

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.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.

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