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Axioms, Postulates, Maps, Territory ….

It began as a response to an interesting comment by Philip Scott Thomas at

http://www.countingcats.com/?p=13915&cpage=1#comment-307104 ,

but I’m afraid it got out of hand…. :(

PST,

Gödel! Indeed. I’m still trying to find a layman’s restatement of The Theorem. Although I believe the theorem says far more, for now I go with “No closed system can explain itself,” by which I mean precisely that every non-trivial logical system must be built on postulates: starting propositions about the system that we assume without proof. (A given person may or may not consider such a postulate self-evident. More on that below.)

You wrote, “the starting point of a system may also be a self-evident statement. I took you to be referring to statements “without which it is impossible to think rationally.” Which, on reflection, I think isn’t what you meant at all. However, that’s going to be my starting point just the same.

However, the description in boldface is vitally important. Such statements are the ones properly called AXIOMS.

Axioms do not in themselves constitute the starting point of any system except the null system–the one having NO system-specific propositions (be they postulates or theorems).

Axioms are about how human logic must proceed. Postulates are starting, hence unprovable, PRE-assumptions about relationships within some particular system.

The axioms are the same whenever we reason logically, regardless of what we’re reasoning about. The postulates are the foundational premises we make about the SPECIFIC system under discussion. It is a POSTULATE that “there is a real world out there which exists whether anyone thinks so or not.”

Here is another foundational postulate: “There is a God, whose properties are ….”
And another: “There exists no ‘God’ whose properties are ….”

We then posit (postulate) a bunch of attributes of the object “God” and see what else we must include in, or exclude from, our set of postulates in order to keep the system logically valid.

Unfortunately there is plenty of precedent for confusing axioms and postulates, as the logicians seem to have adopted a notation and a system of thought where they call all fundamental rules and premises “axioms.” They distinguish between the REAL axioms, which are the rules of thought without which reasoning is simply impossible — such as “A = A” — and postulates, which are the presumptions we’re going to make when building a logical system, by calling the first something like “L-axioms” and the second something like “P-axioms”–I forget the exact labels, but they do refer to both as “axioms” and then add a designator to distinguish between them.

As for “pulling [postulates] out of your arse,” yes, you CAN do that, though you may not end up with anything much–or with anything that seems applicable to the real world. You can start with some collection of elements–numbers, geometric shapes, barbers–and posit, arbitrarily, one or more statements about them (“propositions”). You can then work out the logic. You may start with the set of all barbers, and then postulate that there is a subset of the barbers who shave all those, and only those, who don’t shave themselves. Now that IS a valid logical system. Unfortunately, following out the logic leads to the conclusion that said subset of barbers is the Empty Set. (Russell chose to say that therefore the concept of the subset so-defined is meaningless. I agree with him, but that’s just me. *g*)

With this postulate you have constructed a logical system, albeit one which is inherently trivial. (The system’s elements are “all barbers who shave only those who do not shave themselves. The starting proposition is, “The set of such barbers is not empty.” But the postulate is self-contradictory. This is why Russell called the concept of such barbers meaningless, and it’s why I say we have here a system containing no propositions which are true–thus it is an example of the trivial logical system.)

But some sets of postulates–starting assumptions, givens of the system–which although not intuitively obvious, let alone obviously relating to something in the Real World, yield a rich system chock-full of non-trivial theorems (provable statements about the relationships among various elements of the systems, and about relationships among subsets of the elements of the system).

Such as the starting postulates of Euclidean plane geometry. (E.P.G. may have elements and propositions inspired by the real world, and it may yield results which help us to understand the real world, but its elements, being pure abstractions such as lines with no width, points with no extent at all, etc., have no actual physical referents in the real world. E.P.G. is the map, not the territory.)

When we reason about the Real World, we are trying to develop rules or “maps” that will let us navigate our way to a better understanding of it. So somewhere, there are and must be foundational postulates about that world, one of which is going to be along the lines of “There exists an object X in the real world, which has the property that….”

As a matter of fact, although I’ve never worked out such a system as the very idea bores me, I don’t see why you can’t posit a solipsistic Universe, where all that is exists only in your imagination. I think you could construct a valid logical system around that postulate. I think in fact it would be easy, depending on how nit-picky you are about making unassailable statements–which is a language, or a meta-language, issue.

As for “certain postulates [about the Real World] are self-evident”–well, yes, they are, but the problem is that some people find one postulate self-evident [say the existence of god(s)] and others find the exact opposite self-evident. Now, that does not mean that both those people are correct. Self-evidence may be in the eye of the beholder, but Reality is not: and that is one of MY foundational postulates ABOUT THE REAL WORLD. (It could be fun to write a fantasy novel where that postulate is false. And of course Solipsism is one example of a system that denies the existence of reality external to some particular person’s consciousness.)

Also your own example. It’s self-evident to you and me that all races, say, are human. However, there are two terms in there that need definition: “races” and “human.” The simple fact is that throughout history there have been groups of humanoids who thought that other groups of humanoids were, at best, only somewhat human: subhuman in fact. Famously, a certain large subset of ethnic Europeans held that opinion of Negroes to be self-evident; and those who are rigorously Muslim learn that self-evidently non-Muslims are subhuman. (For instance, only Muslims can, properly speaking, be “innocent.” The concept is simply not applicable to non-Muslims, who can only enjoy full Personhood, at least in current Islamic doctrine, if they convert to Islam. This boils down to denying the full humanity of non-Muslims.)

Nevertheless, it’s self-evident to me personally that all your groups are groups of humans, and also that their members share the same fundamental nature as all other humans…and that therefore they are subject to the same laws of nature, wherefore it is wrong for anyone to assume authority over any other fully-competent (to exclude children and the mentally dysfunctional) human. Which is precisely to say it’s wrong to interfere (unjustly) with another’s self-determination. Which is to say that we all have the same “rights” to life, liberty, and property, by nature.

I’m afraid it seems to me that your penultimate paragraph is circular. :>((

I’m glad if my posting interested you. I’m delighted to see your thoughts on the matter, and I’d look forward to reading more of them.

34 Comments

  1. Julie near Chicago says:

    ‘…[T]here are and must be foundational postulates about [the real] world, one of which is going to be along the lines of “There exists an object X in the real world, which has the property that….” ‘

    Or, more properly, “There exists in the real world an object that we call ‘X,’ which we perceive as having the property [characteristic, attribute, ...] that….”

  2. Mr Ed says:

    ‘human logic’: is there not simply ‘logic’, independent of humans? Logic, like mathematics, is discovered rather than made. Other sentient beings, if they exist or existed, would have the same mathematics, if they got that far, so would not they have the same principles of logic?

  3. NickM says:

    What is the logical value of the statement,

    “The King of France is bald”.

  4. RAB says:

    A business opportunity for Wigmakers?

  5. Mr Ed says:

    I have found Russell to be tedious.

  6. Julie near Chicago says:

    Mr. Ed,

    So I always used to think. As I get older and probably dumber, I also (believe it or not! *g*) tend to grow a little less sure of some of my earlier rock-solid convictions. Namely, it’s no longer entirely clear to me that an alien intelligence would have a “mind” (or stated more properly, from my POV, the kind of physiological structure that would manifest itself as something like “mind”) enough like ours that it would work the same way. Perhaps it would bypass what we think of as “logic” or “mathematics” altogether, and have some other way of dealing with whatever Reality presents to it.

    The human capacity to work with mental pictures of spatial objects seems to exist to varying degrees, so I’ve read, with mathematicians having a better-than-average capacity to do that. Perhaps there’s an alien species that doesn’t do “logic” at all, at least not in the human sense, but “thinks” using “mental” pictures instead, or, heck, sound pictures, or has some other faculty we can’t even imagine.

    As to the nature of Mathematics–does it exist “out there,” in the Wild of Reality, or is it strictly a construction of the human mind? That’s one of the Big Questions and has been for centuries or millenia. Me, I’m almost more inclined to think of Mathematics (and Logic generally) as a faculty of the human mind, a faculty of ours that helps us to cope with Reality, so that Mathematics is properly a matter of both psychology and abstract philosophy.

    A mathematical idea is both a scientific discovery–not about external Reality, but about the workings of the human mind (though not presented in a scientific form, as some physical object like the orbit of Mercury) –and an invention. But because of the omnipresence of Logic in the human mind (flawed as it is in many cases), over the length of time that it’s been recognized however faintly as a field for human consideration and exploration, it has tended to seem like something “out there,” and generally people have come to think of it that way, and treated it that way in statute.

    In short, if there were no mind that could work with the ideas presented by mathematics, there would BE no such thing as mathematics; but, if a tree falls in the forest, it does create a pressure wave in the air whether any entity apprehends that somehow or not. The tree and the pressure wave exist independently of our apprehension of them.*

    (That, of course, is a part of my complex of “self-evident” assumptions. It rests in the end on definitions and postulates in the system of thought–such as it is!–that I’ve built up about the Real World. Others may disagree with me simply because they have a different mental “image” of the world and how it works, with somewhat different definitions and starting assumptions. They may have a better mental representation of Reality than I do, or not. It seems to me they do not, but then again I could, in theory, be wrong ;) ).

    PS.

    *Whether or not you think it makes a “sound” is a matter of your definition of “sound.” Is a sound any air-pressure-wave within a certain range of frequency-wavelengths, or does your concept of it also require a living entity who apprehends it through some faculty of hearing? Depends on your definition.

  7. Plamus says:

    “…but, if a tree falls in the forest, it does create a pressure wave in the air whether any entity apprehends that somehow or not.”

    Or does it? Julie, I assume you are familiar with the double-slit experiment, but if not, do read up on it, or enjoy Feynman’s absolutely awesome presentation of it. Keeping that in mind, is it inconceivable that a tree falling in the forest may create and not create a pressure wave at the same time, and this duality collapses into a pressure wave only when there is an observer? I know, I know, madness (and solipsism) lies that way… but one cannot wish away single-electron interference. Observation changes things in the quantum world… but can our world not be a quantum world to a creature/mind/conscience of galactic magnitude?

  8. Julie near Chicago says:

    Plamus,

    Perhaps you can create a logically consistent theory of reality (or of physics) which describes our observations, on at least the macro-level, as well as the ones most Westerners (at least) nowadays consider common-sensical. If so, have at it! As for solipsism, I’ve said I think it would be relatively easy to construct a valid logical system that starts from the postulate that there is no Reality except what exists in the mind of some guy in Canarsie. *g*

    However, your question (or thought) quotes the last sentence of one of my paragraphs. In answer, let me quote my very next paragraph:

    “That, of course, is a part of my complex of “self-evident” assumptions. It rests in the end on definitions and postulates in the system of thought–such as it is!–that I’ve built up about the Real World. Others may disagree with me simply because they have a different mental “image” of the world and how it works, with somewhat different definitions and starting assumptions. They may have a better mental representation of Reality than I do, or not. It seems to me they do not, but then again I could, in theory, be wrong ;) ).”

    PS. Yes, I’m quite aware of the double-slit experiment. But I didn’t know Dr. Feynman’s presentation is on YouTube–thanks for the link. :)

    PPS. I think it was in one of the short-shorts by SF writer Fredric Brown that God turned out to be a little old geezer in a dusty back-room office, perched on a stool and wearing a green accountant’s eyeshade, who was creating the Universe “in real-time” as he worked out its account in a thick ledger. I suppose there’s no reason why he couldn’t have been working on more than one universe at a time, in more than one ledger. Or indeed, being God, he might even be able to handle the workload of setting out the accounts of infinitely many universes simultaneously. That could explain it, no?

  9. Mr Ed says:

    If a whopping meteorite hits the Earth somewhere remote in say, Russia, does it create a pressure wave or not? It would be, of course, very much a Newtonian phenomenon, but what if the observer is vapourised? :-)

  10. Julie near Chicago says:

    Great Scott, Mr. Ed! I believe you’ve solved the problem of time travel!

    If the observer is vapourised, then the vapourising event must cease to exist at the instant of vapourisation, since it’s only the observer’s observation that permits the pressure wave and hence the vapourisation in the first place. If no observer, then the vapourisation can’t take place. But the incoming meteorite causes the pressure wave, so if there’s no vapourisation there’s no pressure wave, and if there’s no pressure wave there’s no incoming meteorite. But in that case, there was no vapourisation so the observer can’t have been vapourised. So where did he go? He must have gone backward in time at the instant the pressure wave began to contact his noodle, and blown the meteorite off-course or something. That way, when it didn’t hit he could just stand there and not-watch it hitting, so by the theory it wouldn’t hit, and –

    Hm. I hope I haven’t misplaced a decimal point somewhere ….

  11. NickM says:

    Julie,
    I donno about decimal points but you are confusing the hell out of me!

    Yes, the observer in QMech is an issue. Probably intractable – at least to me!

    Of course the many worlds interpretation solves a lot of problems and to my enormous discredit I haven’t read David Deutsch on the issue but it appears he has at least partially solved the issues there. Of course Copenhagen is alive with all it’s oddity. It is the joy of QMech that the orthodox approach is truly bizarre. And so are all the alternatives. The Universe is truly bizarre. And that is cool.

    Typed in my TARDIS.

  12. RAB says:

    Oh it’s all too bind moggling for a simple old rock n roller like me. If I walk into the coffee table and bang my knee, I go ouch! That’s enough reality for me to be going on with. I’m just enjoying the magnificent ride that is existence.

    C S Lewis had something to say about it though (doesn’t he always)

    “If the whole universe has no meaning, we should never have found out that it has no meaning: just as, if there were no light in the universe and therefore no creatures with eyes, we should never know it was dark. Dark would be without meaning.”

    ― C. S. Lewis

  13. Philip Scott Thomas says:

    Julie –

    Thanks for the shout-out. But gosh, the mental gears are creaking. I haven’t had to discuss stuff at this level for years; there isn’t a lot of call for metaphysical speculation in computer programming, you see.

    ‘You wrote, “the starting point of a system may also be a self-evident statement. I took you to be referring to statements “without which it is impossible to think rationally.” Which, on reflection, I think isn’t what you meant at all. However, that’s going to be my starting point just the same.’

    LOL. You’re addressing what I didn’t say? :-)

    When we were taught this stuff, there were two kinds of true statements: those that were self-evidently true and those that were true as a result of previous statements.

    ‘Self-evident’ isn’t the same as ‘obvious’. Like I said, “A ’self-evident statement’ is a statement the opposite of which it is impossible rationally to think.’

    So, in the example I gave, all people, regardless of their differences, share to an equal degree the same nature. Yes, there have been (and are still) people who believe the opposite of that statement. From the Twentieth Century there have even been those, mostly in the social sciences, who denied the existence of such a thing as human nature altogether. But in both cases it is possible, though not without difficulty, to demonstrate why their positions are irrational and contrary to reality.

    Now I have to go lie down. My head hurts. :-)

  14. NickM says:

    The point about the hirsute (or otherwise) nature of the King of France is there isn’t a King of France.

    So in what way is the statement false?

    And it is.

    Of course.

    How about…

    Hamlet was depressed because his uncle killed his father and married his mother?

    Unicorns like to gambol in the woods with comely maidens?

    Gandalf was an Istari?

    Tom Baker was the fourth Doctor?

    Truth is contextual. All those statements are true for a certain value of true.

    Or what about various saints and martyrs who did good things. What if you believe their deeds good but their theological justification dubious?

    And what of Warren Buffet. Bloody good on the stocks and shares but to what extent is he a self-fulling prophecy?

    Reality is not simple. I’m R/G colour blind. But I can paint and stuff. How do you see colour? Often I get it right and often wrong. But I think of colour in terms of hex numbers so I am fine on a computer. You know like FF,FF,FF is full white which looks like the background to this page.

  15. NickM says:

    Oh, and fuck it!

    From the popular song (paraphrased)…

    “Everybody loves baby, baby loves nobody but me.”

    Who is baby?

    And for my next trick there is always the Monty Hall problem!

    Certain to generate a tiswas with maths types but the solution is simples – said the meerkat.

  16. RAB says:

    Oh don’t start with Monty again. It’s a cheat. It boils down to an evens bet.

  17. NickM says:

    It’s not a cheat. It’s probability. Only a cheat if you don’t know it or whatever.

  18. Julie near Chicago says:

    Well, but PST, according to some of the nobs here, if you made a statement in the forest and there was nobody to hear it, then even the pressure wave of the statement didn’t exist, so you didn’t make the statement. So I don’t see why I have to address the statement you didn’t make; I’m only doing it out of the goodness of my heart, see. ;) LOL

    . . .

    Seriously, I have a question–which I’m asking not to be provocative but because I’m interested in the answer. Namely, what is the difference between a statement that is self-evident and one that is obvious?

    Also, as you’ve described it above, it seems to me that you’re distinguishing between theorems (“those [statements] that are true because of previous [true statements]“) and postulates (“statements that are self-evident”), which we assume (in the logical sense) to be true in order to get us started in building a theory. Is that what you mean?

    [Whereas axioms are the bedrock principles of all human logic. (I always say "human logic" because I learned as a little child, my first year in college, that if you just say "logic" with no qualifier, some smart-aleck is going to come back with "how do you know ALL logic works that way? Suppose crocodiles were logical. Would they follow the same principles?")]

  19. Julie near Chicago says:

    RAB–I’m sorry you banged into the coffee table. If you did, you had every right to go “Ouch!” Are you sure you banged into the coffee table? What if the coffee table doesn’t exist? Would banging into it still cause you to go “Ouch!”?

    . . .

    Seriously, I love the C.S. Lewis quote (which is new to me–thank you). And I agree entirely.

  20. Julie near Chicago says:

    Nick–yes, we understand about the King of France’s pilosity problem. (You have read L. Sprague de Camp’s short story “Hyperpilosity,” no?) That is, we understand the reference. I’ve always been suspicious of the logic there, though:

    Suppose the following Statement S is true:

    “There is no X. Therefore for every property/attribute P, the statement ‘X has P’ is false.”

    Let P be a property or attribute, and let Q = not-P, in the sense that Q is the attribute or property of not having P.

    Then, since Statement S is true, the statement “X has Q” is false.

    Therefore X has both property P and property not-P.

    Which is a contradiction.

    Therefore Statement S is false.

    ???

  21. Julie near Chicago says:

    No, I haven’t stated it quite correctly. Should be:

    S = ‘Given that there is no existent X, there is no property P such the “X has P” is true.’

    Let Q = Not-P (Q is the property of not-having-the-property-P). Then both “X has P” and “X has Q=Not-P” are false, which is the same as saying that “both X has not-Q=P and x has not-P=Q are true.”

    In other words, the concept of the nonexistent X as either having OR not-having ANY property is meaningless.

  22. Mr Ed says:

    From Nick’s quote, it was not apparent when the statement re Rex Gaulae* Calvo** was made, so we might ask for more information before commenting on it.

    * declension may be wrong.
    ** substituting Spanish for Latin out of laziness.

  23. John Galt says:

    “And for my next trick there is always the Monty Hall problem!”

    If you start getting the goats out again, I’m leaving.

  24. Sam Duncan says:

    Well, that’s done my head in.

    But Nick’s comment reminded me of an article I think I read in Wired (I’ve certainly lost the link) about research showing that, even in non-colour-blind people, colour perception is linked to language processing. Put simply, if you don’t have a word for a colour, you literally don’t see it. Not only that, but there seems to be an inherent progression of separating out the spectrum that correlates with technological progress: hunter-gatherers see only “light” and “dark”; “blue” creeps in later, then “red”, and so on. Homer probably really did see the sea the same colour as wine.

    And if you’re thinking all this is self-evident – if you don’t have a name for something, you can’t use it – the killer for me was Korea. They have a colour we don’t. Some of us see it as green, some as yellow (it’s a sort of greenish-yellow). But to Koreans, it’s always, clearly, and unambiguously er… whatever they call it, and you must be blind if you think it’s yellow. Think of brown (it’s one of our newest colours); it’s just about imaginable that someone might see it, in certain shades, as red, but some cultures will swear it’s blue. Are they blind? Well, maybe in a sense they are.

    And of course, this all connects with the existence of objective reality. Does colour exist outside our heads? (For me, the electromagnetic spectrum says “yes”, but if we’re unaware of it even though we’re physically equipped to perceive it, it makes you wonder.)

    I wish I could find that link. I actually thought of posting it here at the time, but felt it didn’t really fit.

  25. RAB says:

    Don’t be daft Sam, anything fits on Counting Cats… Post what you like.

    That’s facinating, but I’m having trouble believing it. When I was a child my society may have had a word for a colour, but I as yet didn’t, and I’m pretty damn sure I saw the world the same way then as I do now having learnt all the names for the colours.

    Brown is a new colour, huh? Bleedin hell the Middle Ages were a permanent poo colour surly?

    Now I have heard that certain primitive tribes are numerically blind, having the numbers 1. 2. 3. and then the next number is Lots, but never them being colour blind so to speak.

  26. CountingCats says:

    RAB, now that Sam mentions it I’ve heard the same thing – that blue/brown issue struck a chord.

  27. Mr Ed says:

    Surely most babies see colours before language develops?

    If I were able to register through my eyes the entire electromagnetic spectrum, apart from having a head start at places like this http://www.radarmuseum.co.uk/ in Norfolk, I would receive a lot more information about the world, but as this video from the Royal Institution shows, the limits on colour processing would by set by my brain and the scope of my eyes to see roygbiv.

    http://m.youtube.com/watch?v=iPPYGJjKVco&feature=g-user-u

  28. John Galt says:

    @RAB:

    That was Richard Adams and Watership Down surely?

  29. NickM says:

    Sam,
    Try this one on for size. Corsodyl tooth-paste is what colour? I can’t be bothered going-up to check in the bathroom but I had it wrong until told and now I recognise it like I was daft before. Apart from some oddities like that my R/G has only caused me one problem. The University Air Squadron told me ‘eff off.

    Now here’s another weirdness. A really weird one. You’ve obviously heard that folks can get bonked on the noggin and start seeing again? Or even for the first time. Now there are cases of this where someone blind from birth gets their sight in middle-age and for months they can’t see anything beyond the field they’d have used their stick for. So essentially they could walk perfectly down a street in Manhatten navigating visually fine but if you say, “Hey, that’s the Empire State Building!” they couldn’t see it. They’d fine-tuned their brains.

  30. Andrew Duffin says:

    Godel.

    Cripes.

    I didn’t understand the book, either.

    (GEB of course)

  31. Philip Scott Thomas says:

    Julie

    This conversation is proceeding at the pace of one those chess-by-post games, isn’t it?
    [W]hat is the difference between a statement that is self-evident and one that is obvious?

    An obvious statement is one that is generally accepted as true. It’s ‘common sense’, if you wish. It’s sort of like saying, ‘All men are mortal.’ It’s uncontroversial. True, there are those who believe that Elijah, Moses, Mary and sometimes Mohammed didn’t die, but were assumed bodily into heaven. But that doesn’t mean they weren’t mortal, i.e., able to die. They would say that despite being able able to die, they didn’t.

    A self-evident statement, on the other hand, isn’t necessarily generally accepted. There may be a significant portion of the population who believe the opposite. But the truth-ness of the statement (and the irrationality of its opposite) can be demonstrated by argument.

    Also, as you’ve described it above, it seems to me that you’re distinguishing between theorems (”those [statements] that are true because of previous [true statements]“) and postulates (”statements that are self-evident”), which we assume (in the logical sense) to be true in order to get us started in building a theory. Is that what you mean?

    Yes. Using the definitions you gave us, that is correct.

    What I’ve been puzzling over this weekend is whether mathematical proofs are the only type of proof. I wonder if it’s possible for there to be different type of proof for working out, say, a political or theological philosophy. Not quantitatively different, but qualitatively different.

    We’d be talking about different types of logic, I think. If so, the Goedel might be quite right about mathematical proofs, but not about proofs in other branches of knowledge.

  32. Paul Marks says:

    I love Axioms.

    As for Mr Ed’s question.

    Yes – basic logic is the same for all species. It is independent of them.

    The major philospher ………… (I am afraid that his-her-its name can not be pronounced by a human body – let alone spelt) was agreeing with me on the matter just the other day.

    And this philospher is a hundred feet round blob – who can only see in the X ray part of the spectrum.

    Seriously……

    The idea that there is “Jewish logic” or “Nordic logic” is as absurd as the idea that there is “capitalist logic” and “worker logic”.

  33. Paul Marks says:

    The lack of the second “o” in philosopher is, of course, a sign of my own genius.

  34. Julie near Chicago says:

    Philip,

    Rook to Queen’s four. :)

    (Heh…I do know that the Knight moves in an L-shape, and the Queen RULES!!! Natch, that silly old King is naught but a figurehead, virtually powerless, all but a pawn himself. He desperately NEEDS the powerful Queen to hold him up.)

    . . .

    OK, I think I understand your definition of “obvious.” (Heh…I don’t agree with it, since much that is “obvious” is false. For instance I understand that at one time it was “obvious,” at least to certain segments of the Russian populace, that stones cannot fall from the sky.)

    But of “self-evident statements,” you write:

    …[T]he truth-ness of the statement (and the irrationality of its opposite) can be demonstrated by argument.

    But then you say I am correct in my understanding that your “self-evident” statement is a postulate. I’m not sure what you mean then, because a postulate is by definition an assumption that’s unprovable, that is not demonstrable by argument, within the domain of discourse–i.e., for purposes of the problem under discussion.

    . . .

    Are mathematical proofs the only type of proof? I think it depends on what you mean by “proof.” In the Real World, most assertions or conclusions are considered “proven” if there seems to be a preponderance of weighty evidence in their favor, or if it’s simply “unthinkable” that they be false. Or if they’re the result of direct perception (which, however, is far from reliable–as when people hallucinate, or see a mirage, or experience tintinnitus. And of course in trials, “eye-witness testimony” is famously unreliable–and even more famously, is considered the most persuasive evidence).

    And in the Real World (or any domain of discourse not a subdomain of Formal Logic) we have “induction” to deal with. If every swan I’ve ever seen is white, I’ll probably conclude that therefore necessarily ALL swans are white. Of course you, as a computer type, know that we have to guard against such reasoning when we write programs. *g* “It never occurred to me that this stoopid loop might be endless because the index never does hit up on the BZ instruction–I didn’t think it could go negative without first going to zero!” *sigh* …

    But the single biggest problem, I think, is one of definitions. The language in math is very, very carefully constructed so as to make the definitions as precise as humanly possible, so that within the mathematical subsystem being discussed they are absolutely unambiguous; and we know that any connotations of language-terms in math are to be absolutely ignored.

    For these reasons, proofs in mathematics (or logic) are generally much easier to evaluate for correctness than proofs where words are mostly “fuzzy around the edges,” that is, ambiguous from certain angles, and where connotations do matter quite distinctly, and where new aspects of a situation are forever occurring to us and causing us to re-examine our former conclusions.

    The underlying logic is exactly the same as in mathematics–but the world (i.e. the system) about which we’re arguing is vastly different, presenting many and confusing aspects to us, and the terminology we must use is, as a rule, at least somewhat ambiguous, and is freighted with connotation, which can lead us either toward or away from truly logical answers.

    There’s also the simple fact that in the real world, it can be terribly difficult to know which phenomena are important to our enquiry (and in what sense) and which are not.

    However, as a straight up-down yes-no answer to your question, which amounts to, “is there more than one type of HUMAN logic?” my opinion is, NO. (Not unless you want to extend the concept of “logic,” which properly applies only to the human capacity for reason, to other areas such as æsthetics–the “logic of music” and so forth.)

    Bill Whittle has a very interesting video on how easy it is to exploit the “fuzziness at the edges” of common-language words so as to reach truly outrageous conclusions that seem perfectly “logical.” Worth watching (even though I think all of us are aware of the technique)–he makes it SO clear!

    One other thing. Gödel–I don’t believe he said ALL logically consistent mathematical systems must contain infinitely many true but unprovable statements. Counterexamples are said to include some parts of number theory, and also Euclidean geometry.

    But I would bet that Reality abounds in what we would call “facts” IF we knew of their possible existence as facts and IF we could “prove” them (in the real-world sense).

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