The Opposite of Mendacity...
… is “Truth,” and there are at least two kinds of it. The first involves accessible matters of fact, statements a person can know to be the truth. If a sane person were asked, “Where were you at around 2 PM yesterday,” he or she would almost certainly know the answer. No mystery would shroud the truth. For private reasons the person may choose not to answer truthfully, and depending on the circumstances, the lie may be more or less harmless. If the question were more difficult, like, “Where were you at 2 PM on this date last year,” the most truthful reply may be, “I don’t remember,” the truth of the matter being inaccessible, not because it cannot be known, but simply because it is not known. An infinite number of questions of this sort might be asked, with the answers unknown but knowable. We may call these, questions about the empirical world. They can be answered by taking a look, though sometimes taking the look may be difficult.
The second kind of Truth involves questions that cannot be answered truthfully by even the most powerful empirical look. “Why is it that every effect has a cause?” is a question of that sort. We may say without fear that every effect has a cause, but even if we were to catalog every such instance of causes and effects, we would still not know why it were so that no effect can exist without a cause. We would just know that every effect has a cause.
We approach answers to this sort of question by metaphysical means. We may, for example, imagine a chain of causes and effects progressing back into the past, each cause being the effect of some prior cause or causes. But eventually we would see that such a chain would be endless. We cannot even imagine that the chain would have a beginning. We then might do something like what mathematicians do when they represent an infinite series of, say, prime numbers. They list a few of the numbers separated by commas and then put an ellipsis at the end. When we see “1,2,3,5,7…” we know the three dots mean that the mathematician does not know the last number in the series and, moreover, that he cannot know the number. Such expressions are metaphysical, even though they appear regularly in rigorous mathematics. They transcend the scope of any possible empirical look.
Situations of this sort lead to several “explanations,” none of which completely satisfy. We might say that to ask, “What is the highest prime number,” is not a legitimate question because it has no answer. We may then adopt a metaphysical rule that tells us we should never ask questions of that sort, and while we may not like it that some mathematical fact has restricted our freedom, we would, if we were reasonable people, admit that there was in fact little to be gained by asking unanswerable questions.
But what if the question is changed. Instead of asking “What is the highest prime,” we ask, “Is there no such thing as a highest prime?” In other words, are we asking a legitimate question when we ask if the ellipsis is telling us the truth, that there are in fact an infinite number of primes? A truthful answer is available: yes, it is true – and Euclid himself proved it – there is an infinite number of primes, so the question as rephrased is legitimate. That is, it has an answer.
Immediately after we grasp the infinity of the set of prime numbers, we are led to notice the similarity of the set of primes to the set of causes, which also appears to be infinite. But if we then try to apply something like Euclid’s proof to the set of causes – seeking to prove that it is actually infinite – we see that in order to do so we will have to assign something like numbers to the causes. Well, that can be done. We can map the set of prime numbers onto the set of causes, such that every cause in the infinitely regressive set of causes is identifiable as a prime number. We then apply Euclid’s proof and, presto, we see that, yes, the number of causes is infinite. We have asked a legitimate question, “Is the number of causes infinite?” and have gotten an answer.
But note well, none of this is empirically observable. Just as we do not know the highest prime – and to seek it is illegitimate – so do we have no knowledge of the “first” cause. Why? Because there is no first cause. The word “infinite” means no beginning, no end, so the notion that some cause might be “first” is foolishness.
But then a mathematician more mathematical (and, as it turns out, less metaphysical) than the Mouse might object. “You have used number theory analysis, and Gödel has shown that any system that is subject to number theory analysis is either inconsistent or incomplete.” By “inconsistent” Gödel meant that there was some true theorem derivable from within the system which would be found untrue within the system, and by “incomplete” that there was some theorem derivable from the system that cannot be proven, true or false, within the system. But if we consider each of the prime numbers as a “theorem derivable within the system,” we see that in theory, every one of them can be shown to be truly prime, so the charge of inconsistency could not be effectively brought to bear. But what of incompleteness? Surely it must be the case that no matter how many primes a person was able to list, the list would never be complete, so in a very real sense the system of primes, and consequently of causes and effects, is incomplete. That is, both sets are, by definition infinite so they are both eternally incomplete.
But then the metaphysician, with a bemused look on his face, would answer: “Hmmm. Isn’t that what Euclid proved? That the set of primes is eternally incomplete? And isn’t that what I just proved by applying Euclid’s proof to the set of causes? That it also is eternally incomplete?” To make his idea comprehensible to the mathematician, the Mouse might then put the matter in practical terms. “If there were in fact a Big Bang, then it was not the “first” cause of the universe. It was only the next cause in an infinitely regressive series of causes. Something caused the Big Bang, and something else caused the cause of the Big Bang, etc etc etc …”
The mathematician would, of course, grasp the meaning of the three dots, and would (in the Mouse’s dreams) nod his approval and go forth forever assured that the universe (or whatever) is infinite.
And that is an example of the second sort of Truth. We do not, by that method, know the nature of all the causes that have produced the world (or even a gnat’s eyeball), but we have made a true and definitive statement about the nature of “the all.” We know something for sure that we cannot prove by empirical methods. We have gone beyond the limits of what is scientifically knowable. We have, in a word, said something about “everything” that cannot be grasped by an examination of any of the parts of “everything” or of any finite subset of those parts. The ultimate nature of Nature can only be grasped by methods that do not depend on empirical fact for their validity. They depend simply and only on Reason.
The second kind of Truth involves questions that cannot be answered truthfully by even the most powerful empirical look. “Why is it that every effect has a cause?” is a question of that sort. We may say without fear that every effect has a cause, but even if we were to catalog every such instance of causes and effects, we would still not know why it were so that no effect can exist without a cause. We would just know that every effect has a cause.
We approach answers to this sort of question by metaphysical means. We may, for example, imagine a chain of causes and effects progressing back into the past, each cause being the effect of some prior cause or causes. But eventually we would see that such a chain would be endless. We cannot even imagine that the chain would have a beginning. We then might do something like what mathematicians do when they represent an infinite series of, say, prime numbers. They list a few of the numbers separated by commas and then put an ellipsis at the end. When we see “1,2,3,5,7…” we know the three dots mean that the mathematician does not know the last number in the series and, moreover, that he cannot know the number. Such expressions are metaphysical, even though they appear regularly in rigorous mathematics. They transcend the scope of any possible empirical look.
Situations of this sort lead to several “explanations,” none of which completely satisfy. We might say that to ask, “What is the highest prime number,” is not a legitimate question because it has no answer. We may then adopt a metaphysical rule that tells us we should never ask questions of that sort, and while we may not like it that some mathematical fact has restricted our freedom, we would, if we were reasonable people, admit that there was in fact little to be gained by asking unanswerable questions.
But what if the question is changed. Instead of asking “What is the highest prime,” we ask, “Is there no such thing as a highest prime?” In other words, are we asking a legitimate question when we ask if the ellipsis is telling us the truth, that there are in fact an infinite number of primes? A truthful answer is available: yes, it is true – and Euclid himself proved it – there is an infinite number of primes, so the question as rephrased is legitimate. That is, it has an answer.
Immediately after we grasp the infinity of the set of prime numbers, we are led to notice the similarity of the set of primes to the set of causes, which also appears to be infinite. But if we then try to apply something like Euclid’s proof to the set of causes – seeking to prove that it is actually infinite – we see that in order to do so we will have to assign something like numbers to the causes. Well, that can be done. We can map the set of prime numbers onto the set of causes, such that every cause in the infinitely regressive set of causes is identifiable as a prime number. We then apply Euclid’s proof and, presto, we see that, yes, the number of causes is infinite. We have asked a legitimate question, “Is the number of causes infinite?” and have gotten an answer.
But note well, none of this is empirically observable. Just as we do not know the highest prime – and to seek it is illegitimate – so do we have no knowledge of the “first” cause. Why? Because there is no first cause. The word “infinite” means no beginning, no end, so the notion that some cause might be “first” is foolishness.
But then a mathematician more mathematical (and, as it turns out, less metaphysical) than the Mouse might object. “You have used number theory analysis, and Gödel has shown that any system that is subject to number theory analysis is either inconsistent or incomplete.” By “inconsistent” Gödel meant that there was some true theorem derivable from within the system which would be found untrue within the system, and by “incomplete” that there was some theorem derivable from the system that cannot be proven, true or false, within the system. But if we consider each of the prime numbers as a “theorem derivable within the system,” we see that in theory, every one of them can be shown to be truly prime, so the charge of inconsistency could not be effectively brought to bear. But what of incompleteness? Surely it must be the case that no matter how many primes a person was able to list, the list would never be complete, so in a very real sense the system of primes, and consequently of causes and effects, is incomplete. That is, both sets are, by definition infinite so they are both eternally incomplete.
But then the metaphysician, with a bemused look on his face, would answer: “Hmmm. Isn’t that what Euclid proved? That the set of primes is eternally incomplete? And isn’t that what I just proved by applying Euclid’s proof to the set of causes? That it also is eternally incomplete?” To make his idea comprehensible to the mathematician, the Mouse might then put the matter in practical terms. “If there were in fact a Big Bang, then it was not the “first” cause of the universe. It was only the next cause in an infinitely regressive series of causes. Something caused the Big Bang, and something else caused the cause of the Big Bang, etc etc etc …”
The mathematician would, of course, grasp the meaning of the three dots, and would (in the Mouse’s dreams) nod his approval and go forth forever assured that the universe (or whatever) is infinite.
And that is an example of the second sort of Truth. We do not, by that method, know the nature of all the causes that have produced the world (or even a gnat’s eyeball), but we have made a true and definitive statement about the nature of “the all.” We know something for sure that we cannot prove by empirical methods. We have gone beyond the limits of what is scientifically knowable. We have, in a word, said something about “everything” that cannot be grasped by an examination of any of the parts of “everything” or of any finite subset of those parts. The ultimate nature of Nature can only be grasped by methods that do not depend on empirical fact for their validity. They depend simply and only on Reason.
posted by Benedict S. at 12:17 PM
4 Comments:
mouse,Mathematics is the language of nature.Everything around us can be represented and understood through numbers.If you graph these numbers, patterns emerge. Therefore: There are patterns everywhere in nature.
Neither is any interpretation of mathematics which talks of a beginning of all of Existence to be taken seriously, since the truth of Existence is that there simply *is* no beginning, it being impossible ever to arrive at any point in any Past that can be seen to be a termination of any kind, since cycles govern the picture at the highest levels of Reality and are notable for being continuous. If you attempt to trace matters back to an ultimate beginning, you just go retrogressively through one cycle after another ad infinitum. There *is* no beginning of Existence!
John(A): I take it then that we agree. But how about that sneaky proof?
Are you both sure we even exist? One can say yes, I exist because I have made a comment on this blog but Only the information exists The sender is not who he appears to be,depending on who does the looking.
Robin: I exist therefore I am.
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