“Self-Evident” Does Not Mean “Apparent”

_{Barry Arrington

August 31, 2015

Intelligent Design}

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Many of our materialist friends do not seem to know the difference between the epistemological categories of “self-evident” and “apparent.” I am providing this primer on the difference to help them understand.

Here is a typical exchange where a materialist makes this category error.

Barry: It is self-evident that torturing an infant for pleasure is evil.

Materialist: Yeah, lots of things that have seemed self-evident have turned out to be false. For example, people used to believe it is self-evident that the earth is flat, and they were dead wrong.

Where has M gone wrong? First, M has gone wrong on the basic factual premise of his comparison. The ancients knew the earth was round and even measured its circumference. Great discussion here.

But the fact that materialists continue to spew this factually incorrect chestnut over and over after repeated correction is secondary for our purposes today. More importantly, M has failed to understand the epistemological difference between “apparent” and “self-evident.” “Apparent” means “according to appearances.” M has asserted that it is apparent to many people that the earth is flat. That appearance is false. And by equivocating between “apparent” and “self-evident” he attempts to prove that some self-evident propositions are false.

Nonsense. In the sense we are using it, “self-evident” is not a synonym for “apparent.” Instead, a self-evident proposition is defined as a proposition that is known to be true merely by understanding its meaning without proof. In that sense, is the proposition “the earth is flat” a self-evident proposition? Let’s see.

P1: The earth is flat.

P2: How do you know?

P1: Just go outside and look at it.

What has P1 just done? He has appealed to evidence in order to prove his statement. That very appeal means that his statement cannot be considered self-evident. Go back to our definition. A self-evident claim is one that we know to be true without proof.

An example of a self-evident claim is that 2+2=4. I cannot “prove” that 2+2=4. But does the fact that I cannot prove the proposition mean that I must conclude it is false? Of course not. I know the proposition to be true without proof merely because I understand what it means. Another way of looking at it is that I know for an absolute certain fact that the proposition “2+2 is not 4” is absurd in the sense that it cannot possibly be true, and in order to accept it as true I would have to reject rationality itself.

Unlike the statement “the earth is flat,” the statement 2+2=4 is not merely apparently true, it is necessarily true in any rational universe.

We have a clue that we are not talking about a self-evident truth when a proposition is appended to the word “believe.” Yes, people believe self-evident truths in the sense that they must necessarily accede to the fact that they are true. But people do not “believe” self-evident truths in the sense that they have evaluated the evidence and reached a conclusion they think is justified. Self-evident propositions are not subject to proof or disproof by empirical evidence. They are necessarily true. A person’s belief about a self-evident truth is irrelevant and is therefore rarely expressed. Thus, when one talks about a proposition that is either “believed” or “disbelieved” it is a clue that the proposition is not a proposition of self-evident truth.

This brings me back to my original statement. Numerous materialists with whom I have argued have denied that the statement “torturing an infant for pleasure is evil” is self-evidently true. They always agree that it is true. They never agree that it is self-evidently, necessarily true.

And I always ask them this question: Please describe the circumstances under which the proposition “torturing an infant for pleasure is not evil” is true. I say we can know for an absolute certain fact that the proposition “2+2 is not 4” is absurd because it cannot possibly be true, and in order to accept it as true we would have to reject rationality itself. The same is true of all self-evident propositions. The negation of any self-evident truth is absurd and rationality itself must be rejected in order to accept such a negation. I say the proposition “torturing an infant for pleasure is not evil” is just such an absurd negation of a self-evident truth. You, materialist, say it is not. Please support your assertion.

Dear readers, note that my challenge is extremely risky, epistemologically speaking, because even a single instance where it is met will shatter my project into a million pieces.

Happily, no one has ever come remotely close to answering this challenge. And it is easy to see why.

Comments

SB, If you are not absolutely certain, then you have doubts. I don’t think so. I think that if you aren’t absolutely certain then you can have doubts, but may not. I’m not absolutely, logically certain that the sun will rise tomorrow—but I don’t currently doubt it. This seems like a flexible area of language, so I won’t say that your thinking is wrong. It seems very unlikely that you will agree that disagreement is possible here. So, you are not certain that the part cannot exceed the whole, but if someone asks, you are comfortable saying so. Yes! I’m not absolutely certain that the sun will rise tomorrow, but if someone asks, “LH, will the sun rise tomorrow?” I will say yes. Not “probably” or “it’s always risen in the past, so I surmise…” or anything like that, unless it’s a specialized conversation about probability or certainty. I’ll just say “yes.” I’m comfortable saying the sun will rise tomorrow, even though I can’t be logically perfectly certain of it. There are millions of things that we can misunderstand and be wrong about, but self-evident truths are not among them Your ability to discriminate between “things I can be wrong about” and “things I cannot be wrong about” is suspect. It requires an infallible faculty for discriminating between fallibility and infallibility, for example. And didn’t we establish that it’s possible to be wrong about whether a truth is self-evident? (Else you’d be able to tell me at what value of n n+n=2n ceases being a self-evident truth.) If you can’t tell with certainty whether a given truth is self-evident or not, how do you discriminate with absolute perfection between SETs and suspicions? (At this level, bear in mind we’re talking about the identification of SETs, not the truth of any given SET. After all, how do you know you’re infallible on a question if you can’t tell a SET from a non-SET?) Are you trying to say that since philosophers and scientists have changed their mind about many things, it follows that reasonable people will also change their minds about the laws of thought? Is that your point? If so, it doesn’t follow. So please don’t waste another thousand words to imply that point without making it. First, please relax. Second, not quite. My point is that your self-certainty feels quite reasonable to you—just as a pre-QM thinker would have felt entirely justified feeling that it was indisputable that objects can’t move from point A to distant point C without passing through some intermediary point B. Not just undisputed, but indisputable—it would be irrational to believe that things can move without moving! But that belief was, in fact, wrong. Not because the believers weren’t justified based on their knowledge, but because that knowledge was limited—just as we are all limited. Their belief was equal to your belief in strength and surety. But it was wrong. Why are you infallible when they weren’t? I do also make the point that you’re supposing that you can perfectly perceive the limits of a proposition. A must equal A because there is no counter example, and nothing works without it. This is also equivalent to beliefs that, for example, a particle must be a particle and can’t also be a wave. Those prior beliefs were limited because the believers and their knowledge were limited; they could not predict counterexamples outside their experience and education. Why are you infallible when they weren’t? You know this with absolute certainty, right?. You didn’t say you believe it or suspect it. You characterized it as a fact. You are infallibly sure that there is nothing we can be sure of. Whatever happened to your claim that your mind is “fallible” and that you cannot be perfectly certain about anything?. It’s still there. Do I need to repeat, after every assertion, “but I take the formal position that one cannot be logically certain of anything without an infallible perspective from which to assess it”? As I’ve said elsewhere, I’m comfortable with something like, “I’m certain that I can be certain of nothing but my own uncertainty.” Probably someone else could word that more artfully. But no, I don’t think that any part of my assertions here are infallible. That would be incredibly arrogant. My perception of logical truths is infallible for many reasons. I will list only three: Each one of these arguments presupposes infallibility in order to demonstrate infallibility. It’s a little shocking—how did you not think that we’d see through these? First, I recognize the immediate absurdity of denying them. If the law of identity was not certain, Jupiter could be Saturn. You could be me. I am infallibly certain that Jupiter is not, or cannot be, Saturn. I am infallibly certain that you cannot be me. If the law of non-contradiction was not true, kindness could be cruelty, cowardice could be courage, and, life could be death. I am infallibly certain that these conditions and qualities are incompatible. And what if you’re mistaken about what would happen if the LOI were violated? Your perception of what would happen if the law of identity were “broken” is imperfect. This presupposes, for example, that the law of identity would be broken on a human scale if it weren’t absolute. It could be violated in ways that aren’t apparent to you, and thus not absurd. If a methane molecule on Saturn was also a helium molecule, would you notice? Would it mean you were me and I was you? I don’t think so. As a fallible being, you can never know for certain what would happen if the LOI were different than what you believe it is. Since you can’t be infallibly certain of what the result would be, that result can’t support a conclusion of infallibility, can it? The possibility of error has already crept in. The much greater flaw with this argument is that no part of it explains why the LOI can’t be broken. This is an argument for why you don’t want it to be broken. And I agree, the LOI underpins basically all rational thought. We need it to be true. We assume it to be true. It’s an axiom, not a conclusion—we assume that it’s true because the assumption works, and is important, not because it’s proven. It’s OK to make such assumptions. Second, I can identify and recognize errors and flawed thinking only because there is an unchanging infallible logical standard that exposes them as errors.. I know the difference between a patently true statement and a patently false statement. If someone tells me that I am seven feet tall, I know infallibly that they have made an error because I know the difference between the truth and what was claimed. I am certain that truth is not error.. And what if your perception of the infallible external logical standard is in error? That’s what happened to people who thought a particle couldn’t also be a wave, for example. Granting the existence of an infallible logical standard, that standard’s name isn’t “StephenB.” Whether the standard is fallible isn’t the question—it’s whether you can infallible perceive it. Your example is amusing, but did you think it through? You know your own height because you can use a measuring stick. When we’re talking about axioms, what’s the measuring stick? You can’t measure all cases, to see whether A is literally always A. You assume that it is because you haven’t found a counter-example, and can’t imagine one. But your knowledge and ability to imagine are limited, fallible. You have only those limited faculties to observe and test the axiom. In other words, you can put the measuring stick of yourself up to any axiom, but how do you tell that the measuring stick is accurate? You have none other to use, and it can’t measure itself—if it’s in error, it will measure itself erroneously. Third, I know that my internal logic is perfectly consistent with the logic of the real world. The psychological portion of the law of logic tells me this: If it rains, the streets will get wet.. This fact is perfectly consistent with the laws of nature: when it rains, the streets get wet. I am infallibly certain that my internal logic corresponds perfectly to the logic of the real world. That is why I am also certain that a piece of pizza is less than a whole pie. (I also know that you are infallibly certain of same, which is why I am harsh with you). Oh? What if random Brownian motion evaporates each rain drop before it hits the ground? I agree that we would never expect to see such a thing in the real world; I don’t think, though, that it’s logically impossible. Your example of a logical impossibility is actually logically possible; if you failed to anticipate this condition, perhaps it’s an indication that you aren’t infallible? (I can’t believe I’m trying to persuade someone that he isn’t infallible. I think we’re both equally shocked with each other. I’m proud to remain civil though!) Or maybe you think it’s actually impossible for the rain to not make the street wet. How do you reach that conclusion? By experience of how rain works and reasoning out cause and effect; reason and experience both being limited faculties. As with the other examples, you’re assuming the infallibility of the faculties you’re using in these three approaches, then claiming you’ve used them to show that you have infallible faculties. You’re assuming your conclusion. Do you feel like that’s good logic? Let me ask it this way: how can you show us that you are infallible without assuming that you are infallible? Oh, and by the way, at what value of n does n+n=2n ceases being a self-evident truth? If you don’t know, why not?Learned Hand_{September 6, 2015
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In the sense we are using it, “self-evident” is not a synonym for “apparent.” Instead, a self-evident proposition is defined as a proposition that is known to be true merely by understanding its meaning without proof.
First, I'm not sure what you're referring to when you say "understanding its meaning" and "known to be true". Can you elaborate this? For example, how do you known when you "understand somethings meaning"? This seems equivalent to "know its true nature or purpose" which itself would be a question of truth. For example, one might say once they understand that the ultimate purpose of marriage is to join a man and a woman, then it is self-evident marriage is between a man and a woman. Is that what you're suggesting? But how do you know the ultimate nature of something? Furthermore, this suggests it is marriages "man-ness and woman-ness" that makes it a marriage. That's essentialism. Second, as you pointed out, sometimes things are simple. What you call self-evident truths are just ideas that are very hard to vary, and which we lack good criticism of. For example what would evidence that 2 + 2 = 4 look like? Imagine someone with a box containing two cupcakes adds two more cupcakes but does not end up with four. This scenario indicates that one of our assumptions are incorrect. The question is, which one and why? You will decide it is the the box of cupcakes system does not model two, four and addition. And you will have done so after comparing the two assumptions against each other. What would a good explanation that 2 + 2 does not equal 4 look like? I can't think of one. Why can't I? Because the theory that 2 + 2 equals 4, in reality, is extremely hard to vary without significantly reducing its ability to explain what it purports to explain. Go ahead, try to think of one. This property of being "hard to vary" is why mathematicians mistake it for being self-evident or directly intuited. It is indeed my opinion that 2+2 really does equal 4, so I'm not expecting to find a contrary theory that is nearly as good as an explanation. But this isn't to say that such an explanation could not exit. For example, the hard science fiction book "Dark Integers" explores this very possibility, but for only very large integers. So, I would say there are no special cases of "self-evident" truth. Rather, there are explantations that are harder to vary than others. Comparing them is what we do in practice.
Another way of looking at it is that I know for an absolute certain fact that the proposition “2+2 is not 4” is absurd in the sense that it cannot possibly be true, and in order to accept it as true I would have to reject rationality itself.
If 2 + 2 = 4 really is false, this would imply the operation of laws of physics that would directly interfere with the creation of knowledge in ways we would consider malevolent. Specifically, you'd end up with very bad explanations something along the lines of "there really is no such entity as the number 4 because the proofs of mathematics are profoundly inconsistent and we do not notice because there are laws of physics that act on the neurons in our brains that cause us to unconsciously fill in the gaps in a way that allow us to ignore the physical absence of such entity." So, it's not that we can be absolutely certain that 2 + 2 = 4, but any explanation for why it would be false would itself be a bad explanation. We simply lack a good explanation as to why it would be false. Again, sometimes it's simple. You're making it complicated.Popperian_{September 6, 2015
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Briefly, in response to the claim made that theists believe by religious faith in the immaterial or otherness of mind, my path was quite the opposite. A former dogmatic Agnostic, doubting, as our commenting materialists display here, man's ability to truly know anything, I was caught in a loop of agonizing hyperskepticism and religious insistence that my view was the intelligent view...that theists drew their truths written down by earlier theists...all hobbling around on crutches, furious at we brave nihilists for running about with no apparent rational impediment. I came to a point of mathematical and logical rigor...originally a form of escape from the Sartre-like futile machinations of man. I began to understand the presuppositions behind the most basic of thoughts, the first principles of logic....the necessity of the unmoved mover, a first cause, a place outside of godel's circle. Douglas hofstader, author of "godel, escher, bach", delighted in explaining the mind from the perspective of fractals, of digital and propostional recursions, of sudden unexpected flourishes from mere algorithms. And, while I'll always love him, he only reinforced the necessity of immaterial mind, of irreducible complexity, of an original programmer, a creator. Presupposing mind... creating information, prefiguring causes realized into form. That 2 + 2 = 4 is outside of matter, or better, n + n = 2n. Once again, stressed over and over again in this thread: the self-evident nature of a proposition does not require human understanding. To be sure, many if not most of what is self-evident in a logical system requires a progression of first principles, of necessary truths. I'm not sure BA or SB are denying this fact....especially in light of the irrelevant counter-point thrusts upon this thread. In the above denials of self-evident truths, you mount a defense having borrowed nearly all of your presuppositions from a theistic system of logic only to then deny God's existence...cutting away fervently at the branch upon which you stand. Like a former version of myself, and as some have pointed out, a denial of the ability to be certain about anything is to assert a dogmatic certainty....that one can know for certain that mind is fallible, so it must be comprehensively so...just a chemical collection of maybes...all the while forgetting that sufficient knowledge of the world, not complete omniscience, is possible and practical in making claims of certainty....especially in regards to those first principles...that First Cause. We don't require a catechism to believe in the immaterial mind. One does, however, require an enormous degree of faith to deny the soul and to worship at the feet of Darwin, Huxley, Dawkins, and the like...just mere fallible wads of matter such as they are...mugwump3_{September 4, 2015
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Learned Hand
As a matter of absolute logic, I can’t be perfectly certain of them. As I said in the very beginning, “I don’t doubt it.” Not doubting is not the same thing as being absolutely certain
It is exactly that same thing. If you are not absolutely certain, then you have doubts. If you have no doubts, then you are absolutely certain. You are trying to make them different in an attempt to have it both ways.
You yourself said the sun won’t absolutely certainly rise tomorrow; do you doubt that it will?
Cut it out. I said that I can only know that sun will “probably” rise tomorrow to correct your erroneous claim that you “know” the sun will rise tomorrow. The broader point is that a lack of certainty about future sunrises is not at all comparable to a lack of certainty about first principles, such as the law of whole and parts or the law of non-contradiction.
I was sloppy when I wrote “I’m perfectly comfortable agreeing…”, because that can be read as a statement that I agree that I can be absolutely certain that p/slice can’t exceed p/whole. I didn’t mean that, though, only that if someone asks, “Can the part exceed the whole?”, I would be very comfortable saying “no” and not ever worry about being wrong. I think that’s very clear when you read the whole paragraph.
So, you are not certain that the part cannot exceed the whole, but if someone asks, you are comfortable saying so. You are reasonably certain that a slice of pizza may be less than a whole pizza, but you are a long way from being absolutely certain about it, nevertheless, when pressed, you will say it can’t happen—but when the heat is off, you will reverse your field say that you might be wrong about it after all because your mind is “fallible” and you “cannot know anything perfectly,” In summary, you always try to have it both ways. Is that about it?
Oh, I’m supposed to? Yes, I know I’m supposed to agree with you. But I don’t, and I’ve explained at length why not. And your response, rather than to address those reasons, is literally to complain that I am “supposed to” agree with you. According to who? And how can I agree with you when you won’t actually address the points I’ve made about fallibility? You can repeat “insane,” “liar,” “idiot,” all you like; they don’t add up to an actual argument. I know you feel infallible. How do you know that you actually are? What are the conditions under which it is impossible for you to misunderstand or incompletely understand something?
You are supposed to know that a slice of pizza cannot exceed a whole pizza. You are supposed to know that the laws of non-contradiction and identity are infallibly true and that you can be absolutely certain about it. You are supposed to know that nothing can change or come into existence unless an outside agent causes it to happen. You are supposed to know that it is wrong to slice up babies like pieces of meat while they are still alive and sell them. If you don’t know these things, then you are not a rational person. There are millions of things that we can misunderstand and be wrong about, but self-evident truths are not among them
Ancient philosophers would have felt completely entitled saying, “A particle moving from point A to distant point C must first move through some separate point B.” Or in other words, you can’t get there from here without passing through some kind of middle. But then, physicists upset the apple cart; it turns out that actually there’s reason to doubt that principle under certain conditions.
So what? You want me to respond to these and other points, but you don’t really make a point. Are you trying to say that since philosophers and scientists have changed their mind about many things, it follows that reasonable people will also change their minds about the laws of thought? Is that your point? If so, it doesn’t follow. So please don’t waste another thousand words to imply that point without making it.
An imperfect, limited being cannot know in advance whether there is something they don’t know. We can be so sure of ourselves that we never actually doubt, and we can comfortably live our whole lives assuming the proposition is true, but as a matter of pure logic, we cannot be certain because we are imperfect, limited beings. We cannot know whether there is something we don’t know or are too limited to understand. Even if the principle is perfect and absolute, we aren’t, and we only perceive them with our own minds.
You know this with absolute certainty, right?. You didn’t say you believe it or suspect it. You characterized it as a fact. You are infallibly sure that there is nothing we can be sure of. Whatever happened to your claim that your mind is “fallible” and that you cannot be perfectly certain about anything?.
So I know you have lots of mean things to say about me. Vent! Get it off your chest. But if you don’t mind, and if you’re able, can you also respond to the argument? Assuming you agree that you are fallible, how do you know that your perception of logical truths is infallible? What faculty do you use to conclude that, and how do you know it’s infallible?
My perception of logical truths is infallible for many reasons. I will list only three: First, I recognize the immediate absurdity of denying them. If the law of identity was not certain, Jupiter could be Saturn. You could be me. I am infallibly certain that Jupiter is not, or cannot be, Saturn. I am infallibly certain that you cannot be me. If the law of non-contradiction was not true, kindness could be cruelty, cowardice could be courage, and, life could be death. I am infallibly certain that these conditions and qualities are incompatible. Second, I can identify and recognize errors and flawed thinking only because there is an unchanging infallible logical standard that exposes them as errors.. I know the difference between a patently true statement and a patently false statement. If someone tells me that I am seven feet tall, I know infallibly that they have made an error because I know the difference between the truth and what was claimed. I am certain that truth is not error.. Third, I know that my internal logic is perfectly consistent with the logic of the real world. The psychological portion of the law of logic tells me this: If it rains, the streets will get wet.. This fact is perfectly consistent with the laws of nature: when it rains, the streets get wet. I am infallibly certain that my internal logic corresponds perfectly to the logic of the real world. That is why I am also certain that a piece of pizza is less than a whole pie--every time. No pizza needs to be measured to confirm the point. The law covers them all.StephenB_{September 4, 2015
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LH,
Even if you can’t quite find an actual lie
I say you are an idiot or a liar only if you make idiotic statements or tell lies, like: you are not infallibly sure that a part of a pizza cannot be larger than the whole pizza, or you are not infallibly sure that A=A, or you are logically perfectly certain only that you can’t be logically certain about anything else. Anyone who says any of those things, far less all three as you do, is an idiot or a liar. You insist over and over that you are not a liar. OK.Barry Arrington_{September 3, 2015
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Nice post, LH. When you write,
And if you have infallible beliefs, how do you know they’re infallible–wouldn’t you have to have an infallible faculty for answering that question?
I think I know what Barry believes. I'll make my case, and Barry can correct or confirm. Barry is a theist (possibly a Catholic like StephenB). He believes, by faith, that a part of him is an immaterial spirit or soul that can access, through his God-given rationality, certain infallible beliefs that exist in the "mind of God", so to speak, and are not dependent on experience or on validation from the material world. My best guess. [Added in edit] I'm sure Barry (and we're really speaking about a particular religious worldview here that many hold) recognizes his human fallibility and limitations in many ways, but there is this element of infallibility that arises from the special relationship he has with God through his ability to reason. A quick search on "faith and reason" found this: http://www.catholiceducation.org/en/education/catholic-contributions/fides-et-ratio-faith-and-reason.htmlAleta_{September 3, 2015
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I've written paragraphs upon paragraphs explaining my beliefs. That's not equivocating. I've also answered every question that's been asked of me. Why are you so shy about answering these last questions? My guess is that it's hard, and it's risky. It's hard because you jumped in with a bullish, confident, seemingly easy assertion: 2+2=4 is a SET, and SETs are easy to know! Shut up, doubters, you dummies! But having made assertions like "there is no grey area" and "2+2=4 is a SET," suddenly implications and entailments you didn't predict arose. And those are less fun to deal with. So rather than doing so, it's back to personal insults to keep the conversation easy. I also think that actually discussing the ideas on the table would be risky for you. You've made bold, aggressive assertions of your own infallibility; that's a hell of a thing to support in the face of questions that require careful answers, especially when those answers might themselves have unpredictable implications. Having a serious conversation about ideas means running the risk of being wrong, and that's especially hard when you started out by assuming your own infallibility. But if a conversation is risky, insults aren't--it feels good, and it's easy, and it yanks your beliefs out of the spotlight. But BA, if there's no grey area, what is the greatest value of n for which n+n=2n is a SET? And if someone gets it wrong but believes the answer is right, then isn't it possible to misapprehend a SET? And if you have infallible beliefs, how do you know they're infallible--wouldn't you have to have an infallible faculty for answering that question? Those are hard questions, but it's not like they don't have answers, even from your perspective. But finding them and supporting them would be hard, and risky. "Liar!" is neither. Even if you can't quite find an actual lie, as is evident from your comment at 220.Learned Hand_{September 3, 2015
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LH,
Then you can not doubt, but not be certain.
Only if you equivocate on the word "doubt" or the word "certain," which you delight in doing.Barry Arrington_{September 3, 2015
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LH:
if the situation was reversed, this is the point at which you’d be calling me a liar for pretending to misunderstand.
No, I say you are an idiot or a liar only if you make idiotic statements or tell lies, like: you are not infallibly sure that a part of a pizza cannot be larger than the whole pizza, or you are not infallibly sure that A=A, or you are logically perfectly certain only that you can’t be logically certain about anything else. Anyone who says any of those things, far less all three as you do, is an idiot or a liar. BTW, I am pretty sure you are not an idiot. LH, every time you whine about being called a liar, I will just put up your statements and let the readers judge. I'm happy to do that as many times as you like. The best way to get me to stop pointing out your lies, is to stop telling lies. Barry Arrington_{September 3, 2015
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Aleta,
I used adding velocities as an example. I was quite specific, not fuzzy.
I can't let that go. You say you are a math teacher. So I now know you are very familiar with cardinality and set theory. That means that when you equivocated regarding a proposition applicable to sets to an application that you knew does not involve sets, you were not doing so out of ignorance. You knew exactly what you were doing. And what you were doing was dishonest at its very core. You are so proud of your velocity example. I don't know why, because it reflects very badly on you. Now you can have the last word.Barry Arrington_{September 3, 2015
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As the last word, then, I'll point out that I was not "fuzzy" on 2 + 2 = 4. What I said, multiple times, about an idea Barry doesn't seem to understand, is that when we apply math to the real world, we have to test our models. I used adding velocities as an example. I was quite specific, not fuzzy.Aleta_{September 3, 2015
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Yes, actually, it is. Having no doubt means the same thing as being certain. Another contradiction. You act as if we don’t understand simple English words. Stop it LH. You are embarrassing yourself. Do you doubt that the sun will rise tomorrow? Presumably no. Are you absolutely certain about it, in the same sense in which you’re certain that A=A? Presumably no. Then you can not doubt, but not be certain. Your response only makes sense under one particular definition of “certain,” which in the context of our discussion of absolute logical certainty is pretty obviously not the one I was using. (ETA: In context, I originally said, "Does being self-evidently true mean that something is logically proven, or merely that we have no good reason to doubt it? That’s a serious question, not a rhetorical one. If the latter, then yes. I don’t doubt it, and can’t think of any case or reason that would cause me to." It can be the case both that (a) we're formally uncertain about a proposition, but that (b) we don't have a good reason to doubt it. Pretty standard use of the English language.) As I said to SB, if the situation was reversed, this is the point at which you’d be calling me a liar for pretending to misunderstand. I assume you legitimately didn’t understand.Learned Hand_{September 3, 2015
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LH, there has been a deafening silence from Barry and SB on your repeated question about about whether n+n=2n is a SET for every value of n. I think they would say “no,” based on various comments above. I go into more detail at TSZ; it’s long so I won’t paste it all here. So at some value of n, n+n=2n transitions from a SET to something else. BA says there are no grey areas, and I think they take the position that it also can’t vary from person to person. (Because while two people may vary in their ability to perceive a SET, that doesn’t change whether the SET is a SET.) So n must be some single, discrete number, no matter who’s answering the question. That doesn’t mean that we can actually know what it is, though. They’ve said it’s possible to be uncertain about whether a SET is a SET (I think), so maybe their answer would just be that we can’t know where the line is? It’s some discrete, single number, we just can’t identify it in practice? That would be a clean resolution in part, I think. Except I don’t know how to resolve the false positives problem under their assumptions. What do you do when two people confidently answer 17+17 confidently, without calculating it, but only one gets the right answer? Did one perceive a SET, and the other misperceived it? And the one who got it wrong, was he just acting on intuition or recollection of his training, rather than his SET-sense? That would imply that intuition and acculturation can be subjectively indistinguishable from the perception of a SET. I think that’s true, but I don’t think they agree. I can’t see how either would address the problem. To be fair, it is time-consuming to have these conversations. Maybe we’ll get a more thoughtful engagement from BA later, after COB. SB seems to have time to write, but no inclination to return to these questions, but that might change too.Learned Hand_{September 3, 2015
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Aleta,
I am well-educated layperson who, as a long-time math teacher, has a special interest in the history and philosophy of math.
God help your students. I hope they are smart enough not to fall for your "the whole 2+2=4 is fuzzy" routine. You seem to crave having the last word. OK, the floor is yours.Barry Arrington_{September 3, 2015
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Hi Barry. I'm not angry - amazement is not anger. Also, I'm not an "intellectual", I don't think. I am well-educated layperson who, as a long-time math teacher, has a special interest in the history and philosophy of math. But it seems to me that if you don't understand the issues illuminated by the parallel postulate situation, because you weren't aware of them, then you are not likely to have understood the points I was making in reference to 2 + 2 = 4.Aleta_{September 3, 2015
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LH,
As I said in the very beginning, “I don’t doubt it.” Not doubting is not the same thing as being absolutely certain.
Yes, actually, it is. Having no doubt means the same thing as being certain. Another contradiction. You act as if we don't understand simple English words. Stop it LH. You are embarrassing yourself.Barry Arrington_{September 3, 2015
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zeroseven, I have time to whack only so many moles, and I am certainly not at your beck. And I certainly feel no compunction to whack again a particular mole that I have already whacked two or three times in the thread above just because you are too lazy to read those previous whacks.Barry Arrington_{September 3, 2015
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The principle can be no less certain than the example. It is the former that informs the latter. I don’t think I disagreed with that. As I’ve explained several times, I wouldn’t doubt either principle or example in practice. As a matter of absolute logic, I can’t be perfectly certain of them. As I said in the very beginning, “I don’t doubt it.” Not doubting is not the same thing as being absolutely certain. You yourself said the sun won’t absolutely certainly rise tomorrow; do you doubt that it will? I’ve certainly refined my thinking about my own thinking throughout this conversation, for which I thank you. I don’t think my underlying position has changed, and I don’t think I’ve contradicted any of my positions. If there’s a specific statement you can’t resolve, please let me know and I’ll take a closer look at it. I’m perfectly comfortable agreeing that pep/slice cannot exceed pep/whole. I will never doubt it in practice.
If you accept it in practice and doubt it in principle, then you have contradicted yourself. I was sloppy when I wrote “I’m perfectly comfortable agreeing…”, because that can be read as a statement that I agree that I can be absolutely certain that p/slice can’t exceed p/whole. I didn’t mean that, though, only that if someone asks, “Can the part exceed the whole?”, I would be very comfortable saying “no” and not ever worry about being wrong. I think that’s very clear when you read the whole paragraph. I can’t imagine any circumstances in which I would actually doubt the truth of your examples.
That’s not enough. You are supposed to know that there are no circumstances in which a slice of pizza cannot be greater than the whole pizza. You are supposed to know that that a slice of pizza cannot be greater than the whole pie because the principle says that no part can be greater than any whole. Oh, I’m supposed to? Yes, I know I’m supposed to agree with you. But I don’t, and I’ve explained at length why not. And your response, rather than to address those reasons, is literally to complain that I am “supposed to” agree with you. According to who? And how can I agree with you when you won’t actually address the points I’ve made about fallibility? You can repeat “insane,” “liar,” “idiot,” all you like; they don’t add up to an actual argument. I know you feel infallible. How do you know that you actually are? What are the conditions under which it is impossible for you to misunderstand or incompletely understand something? This seems a lot like my questions about n+n=2n and false positives; when your response is, “You are an insane uneducable elitist worker bee, etc. etc. etc.,” it rather begins to look like you don’t have an answer. (Your comments on ancient philosophy are totally incoherent). For example, if our positions were reversed, this is where you would say, “That’s a lie. You understand. You just don’t want to deal with the ramifications of your position.” I don’t think that you’re a liar, but I do think you’re reluctant to consider these questions—which is why you aren’t doing it. Ancient philosophers would have felt completely entitled saying, “A particle moving from point A to distant point C must first move through some separate point B.” Or in other words, you can’t get there from here without passing through some kind of middle. But then, physicists upset the apple cart; it turns out that actually there’s reason to doubt that principle under certain conditions. Similarly, ancient philosophers would have felt completely entitled saying, “A particle cannot be a wave; it is a particle. It cannot be both.” But then hey presto—those physicists again. It turns out that truth wasn’t so true after all. A philosopher predating modern physics would have no reason whatsoever to doubt those principles. He’d have felt self-sure, confident, and reasonably so. But he would have been wrong. Being human, with a limited and imperfect knowledge of reality, he was unable to predict that there were conditions under which his self-evident truths might not be true. An imperfect, limited being cannot know in advance whether there is something they don’t know. That doesn’t mean that I expect future physicists to upset the “A=A” cart. But what’s the objective, infallible principle dividing “A=A” from “particle=particle”? An imperfect, limited being cannot know in advance whether there is something they don’t know. We can be so sure of ourselves that we never actually doubt, and we can comfortably live our whole lives assuming the proposition is true, but as a matter of pure logic, we cannot be certain because we are imperfect, limited beings. We cannot know whether there is something we don’t know or are too limited to understand. Even if the principle is perfect and absolute, we aren’t, and we only perceive them with our own minds. So I know you have lots of mean things to say about me. Vent! Get it off your chest. But if you don’t mind, and if you’re able, can you also respond to the argument? Assuming you agree that you are fallible, how do you know that your perception of logical truths is infallible? What faculty do you use to conclude that, and how do you know it’s infallible?Learned Hand_{September 3, 2015
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Barry, great, once you have finished whacking moles, I look forward to finding out if 117+117=234 is a SET, and if not, why not.zeroseven_{September 3, 2015
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zeroseven,
Although Barry never admits to uncertainty, his failure to respond to this point clearly demonstrates . . .
that I have time to whack only so many moles.Barry Arrington_{September 3, 2015
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Aleta,
I am also somewhat amazed . . .
You have shown to be a fool, and I understand that makes you angry. You can retaliate by insulting my educational background as much as you like if that makes you feel better. And if you consider yourself an intellectual in math and are nevertheless fuzzy on the whole 2+2=4 thing, you are living proof that George Orwell was right when he said:
Some ideas are so stupid that only intellectuals believe them
Barry Arrington_{September 3, 2015
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LH, there has been a deafening silence from Barry and SB on your repeated question about about whether n+n=2n is a SET for every value of n. Although Barry never admits to uncertainty, his failure to respond to this point clearly demonstrates that he is uncertain about this. He confidently pronounces 2+2=4 to be a SET. But then refuses to explore why it is and how this relates to other mathematical formulas. Sadly revealing, as another contributor to this board might say.zeroseven_{September 3, 2015
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I see. Then I suggest you study the situation as it is instructive. I am also somewhat amazed that I have been trying to discuss the nature of the relationship between math and the real world - someone who seems so absolutely sure about the nature of self-evident mathematical truths - with someone who perhaps doesn't have much of a background in the subject. The parallel postulate story is as basic of a story in the history of math as there is.Aleta_{September 3, 2015
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Aleta,
Barry, I assumed you were familiar with the whole concept
Then your assumption was faulty.Barry Arrington_{September 3, 2015
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SB: One minute you accept the legitimacy of a concrete example, and the next minute you reject the principle that informs it. Learned Hand
This, again, is not a contradiction.
It is most definitely a contradiction. The principle can be no less certain than the example. It is the former that informs the latter.
I’m perfectly comfortable agreeing that pep/slice cannot exceed pep/whole. I will never doubt it in practice.
If you accept it in practice and doubt it in principle, then you have contradicted yourself. The irony is that you don’t even accept it in practice, as indicated in your following statement:
I can’t imagine any circumstances in which I would actually doubt the truth of your examples.
That's not enough. You are supposed to know that there are no circumstances in which a slice of pizza cannot be greater than the whole pizza. You are supposed to know that that a slice of pizza cannot be greater than the whole pie because the principle says that no part can be greater than any whole. I provided the example so that you could understand the principle. You still don’t. You are trying to place one against the other, which is insane. Why you choose to remain uneducable is a mystery. (Your comments on ancient philosophy are totally incoherent).StephenB_{September 3, 2015
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Barry, I assumed you were familiar with the whole concept of the three possible versions of the parallel postulate and the three resulting 2-d geometries. You are, aren't you? This is not a matter of trusting my word on anything - this a very well known story in the history and philosophy of math. See: https://en.wikipedia.org/wiki/Non-Euclidean_geometry#History Added in edit: the most famous conflict in the three systems concerns the sum of the three angles in a triangle, which can equal 180, or be more or less than 180. Did you know that, Barry?Aleta_{September 3, 2015
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Aleta,
How do you explain this, Barry?
I have nothing but your word for the proposition that there is a conflict. And you have no credibility. After all, you are fuzzy on the whole 2+2=4 thing. Maybe there's a paradox, but no one in their right might would trust your word on it.Barry Arrington_{September 3, 2015
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Question for Barry, to try to find out if he understands and/or will acknowledge the difference between a logical system and its application to the real word. There are three different 2-D geometries, depending on which of the three versions of the parallel postulate you adopt. Each of these system is perfectly, absolutely, logically true within itself, even though though they reach conclusions that contradict similar conclusions reached in the other systems. Suppose we have a 2-D surface in the real world. Which of the three geometries apples? There is no logically correct answer to that question. Only by empirical testing could we decide which geometry fits the facts. How do you explain this, Barry?Aleta_{September 3, 2015
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I gather that discussion on the issues is not something you want to do, Barry. I've made many points on some major issues, and you dismiss them all as lies. It is an interesting experience for me to try to discuss something with someone whose world is black-and-white, and who considers himself infallibly correct. But that's OK, because I'm not writing for you anyway. I have benefitted from articulating some of my thoughts, and there may be others who have gotten something out of this.Aleta_{September 3, 2015
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Aleta, It is not circular. But as you demonstrated nicely, that truth, like almost all truths, can be obscured by a determined equivocator.Barry Arrington_{September 3, 2015
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