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 Post subject: The nature of proof
PostPosted: Thu Dec 09, 2004 2:31 am 
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It is literally my job to prove mathematical theorems. Well, it's part of my job; another (around this time of year, much bigger and more taedious) part of my job is examining other persons' attempts at proving particular theorems and judging their success or failure in so doing. In either case, I had better be fairly sure of my ability to distinguish a valid proof from a piece of B.S.

Now, let me distinguish a theorem from a theory. A theorem is a statement in a formal language that is necessarily true, while a theory is a well-supported explanation for observed events. A theorem, once proven to satisfaction, can only be falsified by showing that the language in which it is written is inconsistent; whereas a theory, by its very nature, is falsifiable -- if in some given situation a theory predicts one event but a completely different event is observed to occur, then the theory is invalid.

Here is a theorem, known as the "triangle inequality":
If a and b are non-negative real numbers, then a + b >= sqrt(a^2 + b^2).

Proof. (a + b)^2 = a^2 + a*b + b^2 >= a^2 + b^2 = (sqrt(a^2 + b^2))^2, because as both a and b are non-negative, a*b >= 0. Thus a + b = sqrt((a + b)^2) >= sqrt(a^2 + b^2).

Here is a theory, known as the "ideal gas law":
If P represents the pressure upon a mass in gaseous state, V represents the volume of that gas, n represents the number of molecules in that gas, and T is the absolute temperature of the gas, then P*V = R*n*T for some constant R irrespective of all the other values P, V, n, and T.

The theory is falsifiable by inspection, for if P is held constant and T goes to zero, then V goes to zero as well -- which is simply impossible, because all mass has volume. Nevertheless, it is a good approximation to the real behaviour of gases for values of T significantly greater than zero.

Now, there are many ways to establish the truth of a theorem. Here are a few:
  • Direct proof, as I used above. That is, take the statement of the theorem and show by deductive steps how the premises require the conclusion. In short, if I have a statement like "P implies Q" then I assume P holds and by calculation, show that Q holds as well.
  • Proof by reductio ad absurdum. In this one, if given a statement like "P implies Q", I make the initial assumption of the opposite, that P implies not Q -- but then show that that cannot be the case because some contradiction must then hold if it does.
  • Proof by mathematical induction. Given a statement like "For all n, P implies Q", I first show by some other method that for some n=k P does in fact imply Q, and then using that fact and some way of expanding n from k to other values, I show that P implies Q for all possible n.
  • Proof by counterexample. (This is the one MiB hated so much.) Given a statement like "All P imply Q", if I can find some P for which Q does not hold, the initial statement is disproven. This is important! Really!
  • Proof by case analysis. Given a statement like "P implies Q", I divide P into possible initial sub-cases P_1, P_2, ... P_n, and then show by differing methods that for each sub-case P_i, Q must hold nevertheless.
  • &c.


The common underlying structure of all these proof methods (and all others, indeed) is the use of the strictures of the predicate calculus or some other equivalent formal system as rules for steps in the proof. One identifies the premises and the conclusion of the statement with atoms of the predicate calculus, and by means of mathematical calculation and by logical deduction, the proof is established. If a calculation is done erroneously or a deduction is invalid then the proof as a whole is invalid.

So much for math, though. Math is simple. Even physics is pretty simple. People and their activities are complicated. Let me take for example something said in another place:

Angel on Crack wrote:
There is no good and no evil. If there were, God would have to exist, and absolute morals would have to exist. Without proof of these two requisites, there can be no concept of good and evil except those that are individual.


By the exacting standards I would apply to marking a math exam, this attempt at proof is a total washout. To be fair, though, it might not really be an attempt a proof at all, but rather a sketch of how a proof might go. If the terms were clarified, the statement were made precise, and the logical steps between deductions were explicit (and in the style of the predicate calculus!), it might become a proof that I could accept. I admit now my doubt that such a proof could ever be constructed, but who knows for sure. Besides, whether or not AoC's contention is true is a topic for another discussion entirely.

Now, I fall back on boring old logic as a means for deducing things about the world perhaps merely out of ingrained habit, perhaps indeed instinctually. My brain is my primary organ for processing information, and whatever else it is, it can act like a computer so in a very fundamental way it is a computer.

But, when I step away from the realm of analytic sentences into the much larger realm of observations, I lose the certainty of the a theorem and have to live with the well-foundedness of a theory. Theories are made to unify observations into predictive laws. However, if a theory is truly well-founded, then its predictions will be accurate and treatable as premises in a logical process. Human experience is that well-founded theories do combine with each other logically to make new theories that are also well-founded, though if we are honest with ourselves we test those new theories with new observations just to make sure. In this way, the fact that my brain is a computer (and perhaps more than that, but again, who knows) is useful to help me cope with the outside world.

Many philosophical statements are analytical statements: e.g. "Act as though the maxim of your action should become the maxim of all action." Many are not, but are nevertheless rather similar in form to analytic statements: e.g. "It is erroneous to fail to distinguish the representation of a thing from the thing itself." We are accustomed to combining all these kinds of statements logically in order to deduce new statements, and I believe we are doing things right thereby.

Intuitively -- don't you have trouble understanding how the Christian God can be both three persons and one person at the same time? It's a mystery of the Christian faith, but it's also a logical impossibility. We frown and say, "I believe it is true, though I do not understand it" or "that's utter balderdash" according to our preferences, but we do not say "Sure, I saw Josh the other day. He was there by himself all alone, and his two other selves were there standing beside him at arm's-length from each other." That latter pseudo-sentence is in fact not even valid English: it is semantically ill-constructed and thus at best a meaningless noise and at worst ... pernicious.

Besides -- do you have a better way of making sense of things?


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PostPosted: Fri Dec 10, 2004 9:34 am 
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I hereby give thanks to my logic teacher, for because of him I can say with pride that I understood 100% of what the lady had just wrote.

Threre are two things I would like to contribute to the disscussion:
Logic is basicly an analitic tool we use to help us predict what will happen based on what had happen. Theorys and some theorems contribute to the couse, and because of that they exist. This is basically the utilitarianistic phylosophy that I support.

Another interesting point is the nature of paradoxes. I have encountered two types of paradoxes: the first is Xenon's, which have been solved. Modern math allow endless searies to have a numerical answer. Another type of paradoxes is the self negativism, "This sentence is a lie" is my best example. Those are the only 'real' paradoxes I'm familiar with, and they are coused by the nature of our logic, that doesn't allow anything between 'yeas' and 'no'.

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PostPosted: Fri Dec 10, 2004 10:35 am 
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Now that I am more cognizant than I was previously (and exponentially less bitchy), I can not only defend my apparent potshot at a proof, but finally understand what the hell you were talking about... I must have read this post three times... ah, fuck insomnia. Here we go:

What I did in that statement can be compared to what experienced students do with a math equation or formula that they are readily familiar with: I skipped a few steps. Allow me to backtrack, then we can reevaluate the postulate and see if it still meets the mathematical requirements of a proof, whether I'm right or not, and I'm probably not.


There cannot be a universal concept of good and evil. In order for there to be such an ubiquitous delineation, a determinable standard of what dichotomizes a good act from an evil one would have to exist. Since there is no standing public recourse or model, we derive our criterion from theology and religion. Morality would not exist today if not for religion - humans are the only creatures that have the capacity for language, a prerequisite to organizing, practicing, or being sentient of a larger divinity. Since animals do not share this trait of language, we can use deductive reasoning to determine they have no concept of religion, and since animals are amoral and respond to instinct rather than an ethical code, they have no semblance of morality. (Additionally, all of our fundamental laws of society, and even the attributes we try to ingrain in our children from birth, can, in some way, be derived from religion. Thou Shalt Not Kill, et cetera.)

So, we can agree that morality is a product of religion. If that is the case, then we have to derive our ethical standard from religion, and there must be a being or beings that have the capacity to make that judgement between good and evil, thereby establishing absolute morals.
If there is no absolute standard, we independently and subjectively create our own, and then the concept of good and evil is a "total washout" because everyone is running their individual scheme of it.


A little more verbose, lengthy, pedantic, etc, but I think it makes more logical sense once it's fleshed out and you can see some kind of logical progression.

Personally, I disagree with the premise of making philosophy - what is supposedly an extension of humanities - as, to be brusque, cut and dry as a mathematical proof. Probably because I suck at and loathe math in all forms. The Statistics final today nearly dropped me like a brick. And yes, I think it is true that Kant's categorical imperatives are analytic statements - that is the nature of an imperative. The latter smells like Aristotle, but I can't be sure - even so, he was the "father of logic" and I don't think there's much room for debate that logic and mathematics aren't closely engendered.

It's probably just that I really don't want to believe that my major has anything to do with math.

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 Post subject: Evolutionary
PostPosted: Fri Dec 10, 2004 12:19 pm 
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Actually, I think morals exist without religion. In fact, our most basic morals are the ones practiced by animals.

I'll say that I don't know if a snake has a concept of religion. I'm not going to debate that part. I will say though, a snake's instincts cause it to perform actions we call somewhat moral. Example, poisonous snakes don't bite each other, even when competing for mate. The creatures have an instinct of "thou shalt not kill [your own species]."

Humans, as very social creatures have this taken to another level. Over time, tribes and people that that have embraced the instincts that allow them to reproduce (protect your own members, do not kill them for petty things, etc) prospered. They eventually reached the level of organization to support religious institituions. Those that wish to benefit from being religious members therefore needed some principles that could get people on the bandwagon. The simple answer: use the morals people already posses by vitrue of being the humans that elvolved to be more social creatures.


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PostPosted: Fri Dec 10, 2004 12:47 pm 
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This:
Angel On Crack wrote:
Since animals do not share this trait of language, we can use deductive reasoning to determine they have no concept of religion, and since animals are amoral and respond to instinct rather than an ethical code, they have no semblance of morality. (Additionally, all of our fundamental laws of society, and even the attributes we try to ingrain in our children from birth, can, in some way, be derived from religion. Thou Shalt Not Kill, et cetera.)

In no way leads us tho this conclusion:
Angel On Crack wrote:
So, we can agree that morality is a product of religion.

You're proven correlation, but not causation. What you've basically said is:
paraphrased wrote:
Animals don't have religion. Animals don't have morality. Therefore morality is a product of religion.

It is possible that both morality and religion are a product of some third concept that you've not considered. Or that religion is actually a product of some form of innate morality...

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PostPosted: Fri Dec 10, 2004 1:10 pm 
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Actually, I was separately proving that animals had no sense of religion and that they have no morality, not that because one was true the other was necessarily required. You'll notice I used a parenthetical statement to comment on the human development of the two, and THAT is the wobbly part, if anything is, because of course people will want to argue that our laws come from something other than religion, which you're more than welcome to do if you can prove it. If you can't, you failt to reject my hypothesis as the alternative.

And of course it's possible that religion is a part of an innate morality. It's possible we developed from a rare kind of idiot monkey - it's possible we rose up from the ash. But if you look at the title of the topic, it says something like "The nature of proof."

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PostPosted: Fri Dec 10, 2004 3:37 pm 
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Angel On Crack wrote:
Actually, I was separately proving that animals had no sense of religion and that they have no morality, not that because one was true the other was necessarily required.


Angel On Crack previously wrote:
Morality would not exist today if not for religion...


That's what I was commenting on, what I was suggesting was that it's possible that religion exists because of morality, maybe they exist and perpetuate each other or maybe both of them exist because of a third concept...

Angel On Crack wrote:
You'll notice I used a parenthetical statement to comment on the human development of the two, and THAT is the wobbly part, if anything is, because of course people will want to argue that our laws come from something other than religion, which you're more than welcome to do if you can prove it. If you can't, you failt to reject my hypothesis as the alternative.


This is the statement you're referring too, I presume?

Angel On Crack wrote:
Morality would not exist today if not for religion - humans are the only creatures that have the capacity for language, a prerequisite to organizing, practicing, or being sentient of a larger divinity. Since animals do not share this trait of language, we can use deductive reasoning to determine they have no concept of religion, and since animals are amoral and respond to instinct rather than an ethical code, they have no semblance of morality. (Additionally, all of our fundamental laws of society, and even the attributes we try to ingrain in our children from birth, can, in some way, be derived from religion. Thou Shalt Not Kill, et cetera.)


In that case i'd like to point out some further flaws in the logic of you're argument:

Angel On Crack wrote:
Since animals do not share this trait of language, we can use deductive reasoning to determine they have no concept of religion...


Please explain to me how you managed to deduce that animals have no concept of religion.

Angel On Crack wrote:
...and since animals are amoral and respond to instinct rather than an ethical code, they have no semblance of morality.


Agreed, we can prove that animals respond instinctually to situations.

Angel On Crack wrote:
(Additionally, all of our fundamental laws of society, and even the attributes we try to ingrain in our children from birth, can, in some way, be derived from religion. Thou Shalt Not Kill, et cetera.)


Once again we have corrolation, not causation. Just because such laws can be derived from religion does not prove that they are.

Angel On Crack wrote:
And of course it's possible that religion is a part of an innate morality. It's possible we developed from a rare kind of idiot monkey - it's possible we rose up from the ash. But if you look at the title of the topic, it says something like "The nature of proof."


This is what I believe to be so, whether because it has some evolutionary benefit or whether there is indeed a god, some form of religion does seem innate in use as humans... From the greek gods to modern day treckers people seem to need something greater than themselves to look up to, to emulate or to explain the things we do not yet understand.

I think your whole train of thought is flawed though, if I understand what you're getting at it's that you can't try to prove a philosophy to be true because everything leads back to morality, which stems from religion. Right?

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PostPosted: Sat Dec 11, 2004 4:51 pm 
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<i>This is what I believe to be so, whether because it has some evolutionary benefit or whether there is indeed a god, some form of religion does seem innate in use as humans...</i>

Right, this is a psychological theory that's been around for some time - the human necessity for a strict paternal figure to protect in infancy, we substitute that infantile requirement with something more omniscient and omnipresent. Sure.

<i>From the greek gods to modern day treckers people seem to need something greater than themselves to look up to, to emulate or to explain the things we do not yet understand.</i>

Sure.

<i>I think your whole train of thought is flawed though, if I understand what you're getting at it's that you can't try to prove a philosophy to be true because everything leads back to morality, which stems from religion. Right?</i>

Something like that. The point of fleshing out the argument wasn't to prove it true - the more I thought about the less coherent at the seams it became - the point was to make it a working proof that you could prove or disprove. If anything, my setup and your critique of it leads to disprove Tamayo's original contention - that the "nature of proof", at least in terms of philosophy, religion, morality, and ethics, is as simplistic as an equation. While the world may contain analytic statements regarding facts, that doesn't do anything for something as flexible and as lawless as the nature of human beings.

Seneca: Uhm, okay, poisonous snakes won't bite eachother, but other, larger animals with more cognitive capacity - antelope, for example, wolves, lions, critters WAY higher on the intellectual food chain - WILL kill eachother for food or for mates - things that are actually necessary for survival and proliferation of the species. We fight over extraneous issues such as, for example, one particular topic that has launched more wars and caused more deaths than any other: how about, say, religion? Do animals that EAT THEIR YOUNG seem a little amoral? I mean, I know there was that story about God commanding Abraham to kill his soon, Isaac, to prove himself, but he was like "just kidding" at the end; I didn't think it was like a motif of the thing. If anything, I would argue that religion makes us more petty than anything else.

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PostPosted: Sat Jan 01, 2005 6:27 pm 
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while society insists on combining them, i still personally think that morality and religion (spirituality) should be mutually exclusive.

If religion was birthed from the Big Questions "where do we come from? Why?" how did it end up "God says don't kill people."

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PostPosted: Sun Jan 02, 2005 10:57 am 
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Might I respectfully try to wrest the readers' attention back to my original question? The bumper-sticker aphorism goes, "I have given up my search for truth and am now looking for a good fantasy"; I personally am not quite ready to do that (though some might claim that that is what I am doing, necessarily) but I am extremely interested in the methods whereby one should go looking.

As for example: I have very great respect for intuition. Peoples' hunches about things are very often quite accurate; I know that I trust my own intuition when I wonder about some mathematical problem or whatever. I do not, however, take an intuition about a statement as sufficient cause to deem that statement true or false. For that, I require a formal proof, either to produce a theorem of mathematics or a theory of science. Metaphorically, then, intuition is an useful guide to where to look, but the final path one takes must be paved by logic.

Others look to the procedures and habits of the past. If it worked then, they tell themselves, it will work now. That is not unreasonable, but what is unreasonable are the following three invalid corollaries: (1) the way it worked then is the best way it could possibly work, (2) if the method of the past turns out not to work after all, then there is no way to do it, and (3) if there actually was no way someone did something like this in the past, then there is no way one can do it now.

There are many people who say, "it is true because I am told it is true". I ... do not respect such methods of establishing proof.

I accept that there are many kinds of statements that are true but nevertheless unprovable by the predicate calculus or any other complete formal system such as Russell's. I don't know if the Riemann Hypothesis (that all complex zeroes of the zeta function have real part one-half) is such a statement, for example, but it certainly hasn't been proven true or false yet, so it could be; and knowing the truth or falsehood of that particular statement would be very useful to me. I have to dance around and say, "If the Riemann Hypothesis holds, then ...."

Now, neither I nor anyone else who cares about such things will accept a proof or disproof of a statement like the Riemann Hypothesis that is not valid logically. The Riemann Hypothesis, however, is analytically true or false. Should I be limiting my understanding of real-world events by means of games with symbols? Or is there another, non-logical and yet logically acceptable method of understanding the world?

I don't know of one. I'm altogether willing to allow one to exist; it's logically possible that one should exist. In that there are statements that are true (in the analytical meaning of that word) but not provable by formal means, there might be non-formal means to show their validity. It's just that the phrase "non-formal means of proof" makes my skin crawl, like fingernails on a blackboard.


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PostPosted: Sun Jan 02, 2005 11:38 am 
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Logic, as I've stated before, is simply a certain method of relating a set of presupposed statements to each other in such a way that a new statement is created. All logical statements are what Kant called analytic statements; i.e. they're simply breaking apart a particular concept (or concepts) into its component concepts. The difference between logic as used in mathematics and logic as used in, say, ethics, is that different assumptions are used. Logic cannot "prove" anything without assumptions, and thus logical proof is always conditional: "IF you believe this, THEN, by the nature of the concepts contained in that belief (as well as other assumptions that are often left unstated), you must believe this." Science (and Mathematics, to a certain extent) attempts to reduce the number of assumptions we use to explain things to as few as possible, but assumptions will always be necessary, simply because of the nature of logic. In other words, what I will call "transcendent proof" (proof that is absolute rather than relative to the assumptions made) is impossible.


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PostPosted: Mon Jan 03, 2005 2:02 pm 
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I apologize for the double post, but I just stumbled across a post I made in an earlier thread that would apply to this one quite well.

IcyMonkey wrote:
Before I start explaining my ideas concerning logic, let me clarify something. I adressed this in my first post, but I might as well elaborate upon it, because often this is the biggest problem many people have with my ideas. Particularly in debates with objectivists, (<3 @ Kuh, TGS, MiB, et al.), I am often accused of trying to use logic to "disprove logic". This is absurd. I doubt there's anyone out there who believes that logic is totally useless. Most philosophers who are accused of being "against" logic simply believe that logic and rationality have limits - that they are tools, and like any other tool, they have things they're suited for and things they're not suited for. Reason alone cannot be used to lift a rock, for example. So it's rather obvious reason has limits of some kind. The question is what exactly these limits are. The difference between Ayn Rand and Friedrich Nietzsche isn't that one "likes" rationality and the other "doesn't". Rather, it's that one believes that logic and rationality can work by itself, and can be applied to almost every facet of one's life, and the other believes that it's simply impossible for logic to do this without the support of other things. But I'm getting ahead of myself.

What is logic? Logic is simply the way of relating ideas together to produce new ideas which are useful. Depending upon the context, the usefulness can be expressed in terms of (among other things) correspondence to reality, prediction of future outcome, or discovery of the best method to achieve a desired outcome.

Logic relies upon assumptions. There's no getting around this. Trying to use logic without having assumptions to operate upon is like trying to build a building without any material. The initial assumptions that logic uses cannot be logically proven, because a logical proof of these assumptions must either be based upon those same assumptions, or based upon other assumptions, in which case those assumptions must be proven. Thus, any attempt to ground the assumptions themselves in logic must involve either circular reasoning or infinite regression. Logical assumptions are "alogical", i.e. outside the provability of logic (as opposed to being "illogical", in which case they would be directly contradictory to the what logic would prove).

What this means is that if logic operates upon two different sets of assumptions, it can achieve two different, and perhaps even contradictory, results, and both results would be equally "logical".

In the context of descriptive judgments (i.e. judgments concerning what things exist and how those things exist), our logical assumptions are basically taken from our intuitions. True, we have other assumptions. For example, in elementary-school science we were all taught about atoms; however, at the time we had to basically accept the existence of atoms without really understanding (except in a very vague way) why it makes sense to believe this. However, in these cases we can find out why it makes sense to believe in the existence of atoms. So, in the case of the individual, they're assumptions, but in the case of the entire logical system by which society in general (and the institution of science in particular) operates, they are not. The only assumptions on this scale are, as I've said, human intuitions.

So what are these intuitions? It would be very hard to really catalogue and adequately describe all of them, and even if one could, they would probably seem to be tautologies. For example, one of our intuitions is that we can trust our senses. Not trusting our senses would not be illogical, as some people might believe. There are two reasons one would have for believing distrust of the senses to be illogical: 1) As a direct contradiction to one of our fundamental assumptions, it would seem "wrong". 2) Assuming distrust of the senses would lead to a logical system with conclusions drastically different than the conclusions of the system that is based upon our assumptions.

Any attempt to prove that trusting the senses is more "logical" than not trusting the senses would either a) assume trust of the senses in the proof(*1), or b) Involve deeper assumptions whose denial would cause the exact same problem. I doubt option b is even possible, as I believe that trust of the senses is one of the fundamental assumptions. However, neither case would really solve the problem.

As for applying logic to ethics, things get a bit dicier. Ethics is based upon prescription (what one should do) rather than description (what is). However, there is no way to derive prescription from description, and therefore in the case of ethics our assumptions are composed of not only the intuitions but also the desires. One desire cannot be more logical than another; however, one method of achieving a particular set of desires can certainly be more logical than another. (For a more in-depth explanation of this please consult this thread.)

How do logical descriptions and predictions relate to reality? We cannot know. The problem is, one of our inuitions/assumptions is that our logical desription of the universe is directly correspondent to the state of the universe itself. In fact, the division between objective and subjective reality itself is an intuition/assumption. Any attempt to "logically"(*2) determine whether logic is "objectively" embedded in the universe would itself involve the assumption that this is the case. So the only reason we would have for believing that the universe operates via logical laws is that one of our fundamental assumptions is that it does. Believing otherwise would not necessarily be illogical, if said belief was part of a logical system that did not involve the aforementioned assumption.

(*1) People who argue that not trusting the senses leads to trouble are missing the fact that such "trouble" is only trouble if we're already under the assumption that the senses are accurate. Trusting that the picture of reality presented us by the senses is "correct" is only more practical than not doing so if we base our judgments of practicality on the results of actions which we are observing through our senses.

(*2) "Logically" here meaning using the commonly-accepted logical system - i.e. the one which uses out intuitions as its assumptions.


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PostPosted: Mon Jan 03, 2005 4:54 pm 
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I think the concept of "assumption" needs some elaboration, here. Logic (more precisely, the predicate calculus) is a game with symbols. The game has rules; are those rules what you mean by "assumptions"? If so, that's fair: you're saying that given the particular rules of the game that we are using, then the outcomes will be determined logically; with different rules, we will have different outcomes.

Yes, indeed! The predicate calculus, however, has some very useful properties. It is not inconsistent. If it were inconsistent, then (metaphorically) it wouldn't be a very fun game, because any player of the game could declare victory at any time. It is sufficiently powerful as a language to be able to describe itself -- and furthermore, it is sufficiently powerful to be able to describe many other interesting concepts, such as for example the typeless lambda calculus. With the lambda calculus, we can build almost all of the rest of mathematics.

If we use different rules, we have a different game, yes indeed. (Maybe. It's really hard to find a system of axioms that is consistent and self-describing, but is not equivalent to the predicate calculus.) I'm used to the rules of the predicate calculus, but I acknowledge that they aren't sufficient to prove all possible statements in the language of the predicate calculus. I'm asking you if you have a better way to do things.

But, if by "assumption" you mean "identification of statements in some other language with the atoms of the predicate calculus", then I think you are in error. The atoms of the predicate calculus have no semantics at all. An atom, say "p", is just a mark on the page. We may use the predicate calculus to describe observable events by means of mapping particular events to atoms, but that is our application of the predicate calculus; the mapping itself is necessarily a synthetic action but it does not change the rules of the game.

In that mapping observable events to atoms of the predicate calculus almost always results in useful knowledge after the logical game is played, it's very convenient. It's not entirely surprising; in that our brains can emulate computers, and computers are logical devices, we interpret what we observe of the world logically. What does seem to be surprising to an extent is the frequency with which our interpretations result in accurate predictions. Mapping observable processes to logical atoms as a method of interpretation is quite successful, on the whole.


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PostPosted: Tue Jan 04, 2005 2:28 am 
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Tamayo wrote:
I think the concept of "assumption" needs some elaboration, here. Logic (more precisely, the predicate calculus) is a game with symbols. The game has rules; are those rules what you mean by "assumptions"? If so, that's fair: you're saying that given the particular rules of the game that we are using, then the outcomes will be determined logically; with different rules, we will have different outcomes.


That's part of it, but it goes a bit deeper.

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Yes, indeed! The predicate calculus, however, has some very useful properties. It is not inconsistent.


Ah, but the very notion of consistency or non-contradiction as the goal is one of the fundamental assumptions of our system of logic. Actually, the concept of contradiction itself containts the concept of negation, which is itself something that can only be grasped intuitively -- and that, of course, makes it an assumption. Two statements contradict if one statement is A and the other is not-A, or if one statement contains within it a statement A, whereas the other contains a statement not-A, or if one contains A and the other is not-A, etc.

But what makes something not something else? There is no way to answer that question without assuming, in one's answer, the concept that is being questioned; in other words, there is no way of answering that question that would not be some variation or complication of "A is not-A when A is not-A." What this ultimately boils down to is this: the idea of identity (that a thing is itself -- A is A), and non-identity (which is essentially the same as negation, non-identity referring more to the state while negation refers more to the act of discerning this state) are two of the most fundamental assumptions of our logical system, and must be accepted without question.

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I'm asking you if you have a better way to do things.


And I'm telling you that "better" is relative to the system being used. Mathematical logic is essentially a system created to fulfill certain goals; it can be considered a subset of Human Logic, a group of logical systems which human beings use that share certain (but not all) assumptions in common (the ideas of identity and non-identity being one of those assumptions). The reason they share assumptions in common is that they are all derived from intuitions and judgments that have aided in our survival as a species. (The idea that there is any connection at all between usefulness in keeping us alive and inherent truth is, once again, a byproduct of the very system that was created to help us survive.) Am I saying that our logical system was created by evolutionary necessity rather than being in any way inherent to the universe itself? Yes, I am.

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But, if by "assumption" you mean "identification of statements in some other language with the atoms of the predicate calculus", then I think you are in error.


I don't fully understand what you're talking about here. Keep in mind I don't know much mathematics beyond first-year Calculus. I think you're referring to what are generally called postulates in Geometry, like the definitions of lines and points and such. If that's what you're talking about, then no, those wouldn't be assumptions, they would be definitions -- describing mathematical concepts in a more "traditional" language such as English or French.

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The atoms of the predicate calculus have no semantics at all. An atom, say "p", is just a mark on the page.


Okay, now I'm confused. Are you saying that, say, "line segment A" in geometry describes not some abstract idea, but rather the particular (say) long, thin mound of lead that has been imprinted upon a piece of paper; i.e., that what would traditionally be considered the representation of line segment A is actually line segment A itself? Somehow, I find it hard to beleive that this is what you're talking about. Please clarify so that I can properly address this.

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In that mapping observable events to atoms of the predicate calculus almost always results in useful knowledge after the logical game is played, it's very convenient.


Yes, it is convenient for our purposes... which means absolutely nothing in and of itself.

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It's not entirely surprising; in that our brains can emulate computers, and computers are logical devices, we interpret what we observe of the world logically.


It's not that our brains can emulate computers, it's that computers can emulate our brains. And yes, I realize that you're probably talking about the concept of a computing system more generally, as opposed to the thing I'm typing on. I'd be interested to know, however, whether any computing systems exist outside of pre-sentient organisms, sentient organisms, and things created by sentient organisms. Computers are simply systems that use the same logic that we do. Thus it's not suprising that our evolutionary precursors have computing abilities, and that things we create have computing abilities.

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What does seem to be surprising to an extent is the frequency with which our interpretations result in accurate predictions.


But, see, the predictions themselves are only correct when analyzed and processed using the same rules that generated the prediction in the first place.


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PostPosted: Tue Jan 04, 2005 12:44 pm 
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Here is what is meant by "atom", in context. I use a high-level construct, namely a grammar fragment, to explain the very-low-level idea of an atom because even though the predicate calculus can be used to encompass itself, it's a pain in the neck to do it.

Code:
proposition      ::= atom
                  |  "~" proposition
                  |  "(" proposition binop proposition ")"

atom             ::= "p" primes
                  |  "t"
                  |  "f"

primes           ::=
                  |  "'" primes

binop            ::= "&"
                  |  "|"
                  |  "->"


In other words, an atom is either a "t" or an "f", or else a "p" with zero or more single-quote characters attached to it. Those letters might be indicative of semantic content, but that fact is merely coincidental.

In many (okay, nearly all) cases, we abuse notation by creating new symbols to represent the unwieldy "p" with the string of single-quote characters. Instead of using the string "p'''''''''''''''''''''''", for example, it's far easier to name that string "a". It is understood, however, that "a" only represents something like "p'''''''''''''''''''''''".

Now, one of the rewrite rules of the predicate calculus goes like this:

Code:
~~A => A


The rewrite rules are the formal rules of the game. If I have a string like "~(~p&~~~p')" then I am allowed by the rules to write the string "~(~p&~p')". (Another rule allows me to transform that one into "(p|p')".) I'm not assigning meanings to anything. I'm metaphorically scribbling symbols onto a piece of paper.

When first we learn logic, we pronounce the symbols "~", "&", and "|" respectively "not", "and" and "or". That is because the semantics of those words seems to map very well to those symbols in the predicate calculus. When we are more experienced with logic, we continue to pronounce those symbols thusly because we are used to doing so, but we know that the symbols hold no such meaning. They are merely scribbles.

Now, you quibbled about my saying that a brain could emulate a computer, and suggested it was the other way around. No, I meant what I said. A person can follow any algorithm, because a person can follow the algorithm for a universal Turing machine. The purest expression of an algorithm is as a Turing machine. Anything a general-purpose computer can do, a universal Turing machine can do, so a universal Turing machine is a general-purpose computer. A computer can execute any Turing machine as an algorithm. So, indeed, a person is a computer -- though it is arguable whether or not a person is more than a computer.

The concept of contradiction is another of the rewrite rules of logic. That is, for any predicate A,

Code:
(A&~A) => f


That is, a contradiction is like a losing move in a game. Any interesting game has to have winners and losers; if we could accept contradictions, then we could accept any well-formed predicate. Again, there is no semantic content to the above rewrite rule: it's just a way of transforming strings in a particular language into other strings in the same language.

I still don't know what you mean by "assumption", though.


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