>So this question has developed from one of epistemology to one of history of philosophy.

Not at all. Do you seriously think the various problems of epistemology were just there when Ayn Rand woke up in the morning? Obviously she didn't - she addressed her replies, no matter how vaguely or disparagingly, to various prior thinkers. They are questions with a history, not questions about history!

It seems to me you believe Rand has the all answers in a field like mathematical epistemology but are none too clear on what the questions are.

It seems to me you believe Rand has the all answers in a field like mathematical epistemology but are none too clear on what the questions are.

Dan,

I know Rand's theory fairly well, and I don't think she made any comment on mathematics as such, except the job of counting, which someone recently said was foundational, rather than mathematical. That's how I view Objectivism as a whole - foundational - especially in speculative metaphysics.

Claiming that math expressions are analytical, denying the validity of the senses, attacking Rand's theory of mind or concept formation... it all seems so pointless. What are you after? Capitulation?

>Claiming that math expressions are analytical, denying the validity of the senses, attacking Rand's theory of mind or concept formation... it all seems so pointless. What are you after? Capitulation?

Errr...truth?

And I might add:

1) It is perfectly reasonable to describe math expressions as analytical. It is a handy distinction, and it matters little if it breaks donw in some circumstances. I am not sure why you are picking this bone with me, it's not a subject I have pursued with much vigour at all on this thread. Perhaps you are thinking of someone else?

2) I don't deny the validity of the senses, only their supposed infallibility. Where did I say this?

3) Rand's theory of mind and concept formation seem fraught with obvious blunders to me. You are welcome to dispute my criticisms of course, which may all be completely mistaken. If you're right, and it turns out that Rand's theories are robust, so much the better. There is a short laundry list at post #176.

I have a better suggestion than the approach so far. If you have a specific case in mind where you think a math construction has no connection at all with reality, or one which did not before and did later (and your "Platonic suspicions" for why that happened), why not present it?

All that has gone on so far about this issue is a bunch of nothing. You have asked questions and I have responded to the best of my ability while you have said that they were not the answers you wanted and have constantly modified the question. I asked only one question and got rhetoric as an answer, i.e. I asked "which problems" and you said "the same problems" as if this were a description of them.

Let's talk about an actual idea. That other stuff is boring.

>Let's talk about an actual idea. That other stuff is boring.

If you were really interested in the actual ideas behind mathematical epistemology, Mike, I think you would have read or genuinely tried to learn at least something about the subject at some stage in your life. But it's quite clear you haven't, so I can only assume that at bottom you aren't all that interested. This lack of both knowledge and interest, however, does not seem to stop you holding some strong and even highly dismissive opinions on the subject. Who knows why you do, but you do. So given all that, I'll sit this one out, thanks anyway.

Claiming that math expressions are analytical, denying the validity of the senses, attacking Rand's theory of mind or concept formation... it all seems so pointless. What are you after? Capitulation?

Just curious,

Wolf

There comes a point in most of these arguments where one has to defend Rand, or defend (or accept) the truth. The journey and the results are most interesting on many levels.

There comes a point in most of these arguments where one has to defend Rand, or defend (or accept) the truth.

When I got home recently after a five month tour, I saw that the back gate was weathered and weakened beyond repair. It had to be replaced, to keep the dog in, the neighbor's chickens out, yet facilitate access to the orchard. Fencing it shut was not a good option. So, I went to the hardware store and bought sufficient material to build a new gate. I used my eyes, ears, tools, fingertips, tape measure, knowledge of fasteners, a little math and a lot of prior experience. The terms used in this statement of history are conceptual. I can define them.

Objectivism is a philosophy for living on earth, whether it pertains to math, validity of the senses, etc.

Claiming that math expressions are analytical, denying the validity of the senses, attacking Rand's theory of mind or concept formation... it all seems so pointless. What are you after? Capitulation?

Just curious,

Wolf

There comes a point in most of these arguments where one has to defend Rand, or defend (or accept) the truth. The journey and the results are most interesting on many levels.

Bob

Aristotle once said -- I love Plato but I love the truth even more.

There is one obvious difference between mathematics and natural language. In mathematics definitions include ALL particulars and so deductions work absolutely, if correctly made. In natural languages there are always characteristics present that are not accounted for and so deductions can only ever work relatively well. Not exactly sure how this relates to the analytic-synthetic debate but it seems important.

I am glad to hear someone praise Aristotelian logic as compared to modern logic: modern logic has done a terrible job regarding conditionals. And this spills over into their account of the predicate logic of universals.

Not true. First Order Logic (Predicate Logic) is the main logic of mathematics. There is nothing wrong with it.

FOL and Set Theory (purged of its contradictions) is basically all that is needed to ground correct mathematical arguments (proofs).

For other than alethic modalities (true/false) one must extend logic to handle other possibilities.

If you have a specific case in mind where you think a math construction has no connection at all with reality, or one which did not before and did later (and your "Platonic suspicions" for why that happened), why not present it?

Michael, about 12 years ago, I was taking part in a David Kelley-led online seminar, and at one point, the question was raised about "what in reality" do zero exponents (x to the zero power) refer to? Because, as you know, good Objectivists always subject their ideas to "Rand's Question": what in reality gives rise to the necessity for this concept? Here is what I wrote and posted to the group:

In December of 1995, while participating in the Institute for Objectivist Studies' cyberseminar on the nature of propositions, it occurred to me that there was a (fairly) simple explanation for the nature of zero exponents. Many people seem mystified by the mathematical idea that any number to the "zero" power is equal to 1. No matter how small or large, this is true. 5-to-the-0-power, nine-million-to-the-zero-power, 3/5-to-the-zero-power...you name it, and it's equal to 1. Why should this be so?

Consider positive exponents: 5-squared (five-to-the-second-power or 5x5) is equal to 25 because (we are told by our teachers) the square (exponent of two) means that you have two factors of the number 5 (i.e. you multiply 5 times itself). Thus, 5-to-the-fourth-power would mean 5x5x5x5 (or 625).

Negative exponents are a little trickier: 5-to-the-negative-second-power is equal to 1/25 because a negative power is (supposedly) defined as the exponent of a number that is the denominator of a fraction with a numerator of 1. In this case, 5-to-the-negative-second-power is equal to 1 divided by 5-squared, which is 1 divided by 25, which is 1/25.

But what possible meaning can there be to multiplying a number by itself zero times? There is no such mathematical operation, is there? And even if there were, why should the result always be 1??

Yet, when we are taught to multiply by adding exponents, we have to do it that way so that the math will come out right. For instance: 6-squared times 6-squared equals 6-to-the-fourth by the exponent rule, and (6x6)x(6x6) = 36x36 = 1296, which is 6-to-the-fourth. No problem there. And: 6-to-the-zero times 6-squared equals 6-squared by the exponent rule, and 1x(6x6) = 1x36 = 36, which is 6-squared.

So, what is the deal with zero exponents? Why do they work? Are they just a "convenient fiction"--something to help the math "come out right"? Or is there something that our math teachers aren't telling us?

The confusion comes in the way that exponents are described. The basic function of an exponent is to relate the operations of multiplication and division not (primarily) to the number of which it is an exponent, but to the unit number, 1. (This, of course, is the basis of the much-dreaded "scientific notation," which I think is how students should be taught exponents, rather than the more colloquial "times-itself" approach.)

This relation to the unit 1 is explicit for negative exponents: x-to-the-minus-3rd-power, for example, is 1 divided by x-to-the-3rd-power. For positive exponents, it is left implicit (since 1 is the multiplicative identity), but is more fully stated, for example, x-to-the-3rd-power is 1 times x-to-the-3rd-power. (x-to-the-3rd = 1 times x-to-the-3rd) .

In light of this, a zero exponent's non-fictional relation to the unit 1 becomes clearly plausible. Here's how it works:

For any real number, r, a positive exponent, n, indicates that the unit 1 is to be considered as having been multiplied by r a total of n times. A negative exponent, -n, indicates that the unit 1 is considered as having been divided by r a total of n times.

A zero exponent, by contrast, indicates that the unit 1 is considered out of any multiplicative or divisive relationship to the base number r. We are to consider only the unit 1 and to refrain from combining it (by multiplication or division) with the exponent's base number.

Now we can see the root of the confusion inherent in the way most of us are taught about zero exponents. The unit 1 should be considered not as being multiplied (or divided) by some base number "zero times," but as itself, i.e., as not having been multiplied (or divided) by the base number any times.

I apologize to the mathematicians in the group for the inelegance of this explanation. I hope it is clear enough for everyone else.

Interestingly, several years ago, Leonard Peikoff announced in a QA session that one of his "brilliant" students had reached a discovery/insight similar to the one I gave in 1995 and posted on my website shortly after. I guess it was just "in the air" or something...

I am doubtful that counting is applied math. It doesn't seem to apply any of the branches of math, even arithmetic. It is seems rather to be an assigning of numerals (which are symbols--verbal or non-verbal, written, spoken or otherwise--that stand for numbers) to things, as part of a procedure in which the last number counted is what we say is the number of the things we were counting. It is more of foundation for math than an application of it.

The operation of counting is part of what cardinality is. Set theory had to be invented (by Cantor and others) to produce a complete definition of cardinality and ordinality.

Ba'al Chatzaf

But that is not a foundation for counting: we understood how to count long before we had set theory.

I am glad to hear someone praise Aristotelian logic as compared to modern logic: modern logic has done a terrible job regarding conditionals. And this spills over into their account of the predicate logic of universals.

Not true. First Order Logic (Predicate Logic) is the main logic of mathematics. There is nothing wrong with it.

FOL and Set Theory (purged of its contradictions) is basically all that is needed to ground correct mathematical arguments (proofs).

For other than alethic modalities (true/false) one must extend logic to handle other possibilities.

Ba'al Chatzaf

Modern Predicate Logic interprets universal generalizations as conditionals (which is good when dealing with kinds), and conditionals are part of Propositionsl Logic, and modern Propositional Logic, in Russell's version, mishandles the conditional, interpreting it is a material conditional, and so that affects Predicate Logic.

I'm not sure what you last sentence means. 'Alethic' does literally mean having to do with truth, but unfortunately 'alethic modalities' is used to refer to necessity, contingency, possibility, impossibility, actuality, etc.

I apologize to the mathematicians in the group for the inelegance of this explanation. I hope it is clear enough for everyone else.

Apology accepted. It's really quite simple. Let x and y be any real numbers, x not 0. (x^y)/(x^y) = x^(y-y) = x^0. Clearly (x^y)/(x^y)= 1. Hence, x^0 = 1.

I am doubtful that counting is applied math. It doesn't seem to apply any of the branches of math, even arithmetic. It is seems rather to be an assigning of numerals (which are symbols--verbal or non-verbal, written, spoken or otherwise--that stand for numbers) to things, as part of a procedure in which the last number counted is what we say is the number of the things we were counting. It is more of foundation for math than an application of it.

The operation of counting is part of what cardinality is. Set theory had to be invented (by Cantor and others) to produce a complete definition of cardinality and ordinality.

Ba'al Chatzaf

But that is not a foundation for counting: we understood how to count long before we had set theory.

The essence of counting is setting up a one to one correspondence between two sets. One set is the set of integers between 1 and n for some n. The other set is, say for example, the set of cows in your pasture. That is how you count the cows.

In days of yore we counted by matching a pile of pebbles to whatever it is we needed to count. Or we used our fingers for small numbers.

Representing large numbers by means of a numerical notation happened much later. The first attempts at that was to commandeer the alphabet. Later on we used the abacus and then positional notation.

Let x and y be any real numbers, x not 0. (x^y)/(x^y) = x^(y-y) = x^0. Clearly (x^y)/(x^y)= 1. Hence, x^0 = 1.

Merlin,

I had to look up lattices to understand this and things started spinning. I am curious about something. I read that the symbol ^ means "and" between true symbols for the group to be true. So if I take your equation and spell it out in words, but let x=6 and y=7 (and presuming that both 6 and 7 are true), here is what I come up with:

The true group of 6 and 7 divided by the true group of 6 and 7 equals the true group of 6 and the subgroup of 7 minus 7, and this equals the true group of 6 and 0. Clearly the true group of 6 and 7 divided by the true group of 6 and 7 equals 1. Hence, the true group of 6 and 0 equals 1.

I'm not sure what you last sentence means. 'Alethic' does literally mean having to do with truth, but unfortunately 'alethic modalities' is used to refer to necessity, contingency, possibility, impossibility, actuality, etc.

There are other modalities besides truth, falsity, possibility and necessity. There are the epistemological modalities such as known/unknown and temporal modalities such as sometime, always, eventually and never. Then there are deontic modalities such as permitted/forbidden/optional. There are modalities such as likely, probably. There are is believable/unbelievable, the modalities relating to plausibility of propositions.

Unfortunately once you get beyond the the modalities of true/false the expressions are no longer easily analyzable in terms of their components. In short, no truth tables and such like contrivances.

There are other extensions of logic. One interesting extension has to do with -entailment- where there has to be a semantic connection between the premises and the conclusion. With material implication we get non-sense such as "the moon is made of green cheese" implies "George Bush is the president". Unfortunately, under material implication a false statement implies any statement. Entailment addresses the semantic content of the components of a conditional. See http://en.wikipedia.org/wiki/Entailment and the references therefrom.

Now we can see the root of the confusion inherent in the way most of us are taught about zero exponents. The unit 1 should be considered not as being multiplied (or divided) by some base number "zero times," but as itself, i.e., as not having been multiplied (or divided) by the base number any times.

I apologize to the mathematicians in the group for the inelegance of this explanation. I hope it is clear enough for everyone else.

Here is another 'intuitive' way of looking at it, take a number like 123.45

In positional notation (base 10) this number breaks down to this.

1x10^2 + 2x10^1 + 3x10^0 + 4x10^(-1) + 5x10^(-2)

So you see it's right there in front of us, 3x10^0= 3 therefore 10^0 =1

Of course this doesn't prove x^0=1 for all x but it does give one a good idea about why it must be so, I think.

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## Daniel Barnes

Wolf:

>I admire your patience.

Haven't you got some important concepts you need to go off and define or something?

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## Daniel Barnes

Mike:

>So this question has developed from one of epistemology to one of history of philosophy.

Not at all. Do you seriously think the various problems of epistemology were just there when Ayn Rand woke up in the morning? Obviously

shedidn't - she addressed her replies, no matter how vaguely or disparagingly, to various prior thinkers.They are questions with a history, not questions about history!It seems to me you believe Rand has the all answers in a field like mathematical epistemology but are none too clear on what the questions are.

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## Wolf DeVoon

Dan,

I know Rand's theory fairly well, and I don't think she made any comment on mathematics as such, except the job of counting, which someone recently said was foundational, rather than mathematical. That's how I view Objectivism as a whole - foundational - especially in speculative metaphysics.

Claiming that math expressions are analytical, denying the validity of the senses, attacking Rand's theory of mind or concept formation... it all seems so pointless. What are you after? Capitulation?

Just curious,

Wolf

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## Daniel Barnes

Wolf:

>Claiming that math expressions are analytical, denying the validity of the senses, attacking Rand's theory of mind or concept formation... it all seems so pointless. What are you after? Capitulation?

Errr...truth?

And I might add:

1) It is perfectly reasonable to describe math expressions as analytical. It is a handy distinction, and it matters little if it breaks donw in some circumstances. I am not sure why you are picking this bone with me, it's not a subject I have pursued with much vigour at all on this thread. Perhaps you are thinking of someone else?

2) I don't deny the validity of the senses, only their supposed

infallibility. Where did I say this?3) Rand's theory of mind and concept formation seem fraught with obvious blunders to me. You are welcome to dispute my criticisms of course, which may all be completely mistaken. If you're right, and it turns out that Rand's theories are robust, so much the better. There is a short laundry list at post #176.

Edited by Daniel Barnes## Link to comment

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## Michael Stuart Kelly

Daniel,

Yes. Her and everybody.

Was there ever a time when there was no seeking of knowledge about thinking? That's all epistemology is.

Michael

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## Daniel Barnes

Mike:

>Was there ever a time when there was no seeking of knowledge about thinking? That's all epistemology is.

Thanks for clearing that up.

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## Michael Stuart Kelly

Daniel,

I have a better suggestion than the approach so far. If you have a specific case in mind where you think a math construction has no connection at all with reality, or one which did not before and did later (and your "Platonic suspicions" for why that happened), why not present it?

All that has gone on so far about this issue is a bunch of nothing. You have asked questions and I have responded to the best of my ability while you have said that they were not the answers you wanted and have constantly modified the question. I asked only one question and got rhetoric as an answer, i.e. I asked "which problems" and you said "the same problems" as if this were a description of them.

Let's talk about an actual idea. That other stuff is boring.

Michael

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## Daniel Barnes

Mike:

>Let's talk about an actual idea. That other stuff is boring.

If you were

reallyinterested in the actual ideas behind mathematical epistemology, Mike, I think you would have read or genuinely tried to learn at leastsomethingabout the subject at some stage in your life. But it's quite clear you haven't, so I can only assume that at bottom you aren't all that interested. This lack of both knowledge and interest, however, does not seem to stop you holding some strong and even highly dismissive opinions on the subject. Who knows why you do, but you do. So given all that, I'll sit this one out, thanks anyway.## Link to comment

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## bmacwilliam

There comes a point in most of these arguments where one has to defend Rand, or defend (or accept) the truth. The journey and the results are most interesting on many levels.

Bob

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## Wolf DeVoon

When I got home recently after a five month tour, I saw that the back gate was weathered and weakened beyond repair. It had to be replaced, to keep the dog in, the neighbor's chickens out, yet facilitate access to the orchard. Fencing it shut was not a good option. So, I went to the hardware store and bought sufficient material to build a new gate. I used my eyes, ears, tools, fingertips, tape measure, knowledge of fasteners, a little math and a lot of prior experience. The terms used in this statement of history are conceptual. I can define them.

Objectivism is a philosophy for living on earth, whether it pertains to math, validity of the senses, etc.

W.

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## BaalChatzaf

Aristotle once said -- I love Plato but I love the truth even more.

Ba'al Chatzaf

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## tjohnson

There is one obvious difference between mathematics and natural language. In mathematics definitions include ALL particulars and so deductions work absolutely, if correctly made. In natural languages there are always characteristics present that are not accounted for and so deductions can only ever work relatively well. Not exactly sure how this relates to the analytic-synthetic debate but it seems important.

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## BaalChatzaf

Not true. First Order Logic (Predicate Logic) is the main logic of mathematics. There is nothing wrong with it.

FOL and Set Theory (purged of its contradictions) is basically all that is needed to ground correct mathematical arguments (proofs).

For other than alethic modalities (true/false) one must extend logic to handle other possibilities.

Ba'al Chatzaf

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## Michael Stuart Kelly

Bob,

That's too broad to be anything but a personal opinion of Rand as a whole. The implication is that Rand's ideas are false about everything.

I know my reaction is not one of intellectual engagement with something like that. I think, "Well, the guy doesn't like Rand." And I stop there.

Michael

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## Roger Bissell

Michael, about 12 years ago, I was taking part in a David Kelley-led online seminar, and at one point, the question was raised about "what in reality" do zero exponents (x to the zero power) refer to? Because, as you know, good Objectivists always subject their ideas to "Rand's Question": what in reality gives rise to the necessity for this concept? Here is what I wrote and posted to the group:

================================================================

What's the Deal with X-to-the-Zero-Power??

A Brief Note on the Non-Fiction of Zero Exponents

by Roger E. Bissell

In December of 1995, while participating in the Institute for Objectivist Studies' cyberseminar on the nature of propositions, it occurred to me that there was a (fairly) simple explanation for the nature of zero exponents. Many people seem mystified by the mathematical idea that any number to the "zero" power is equal to 1. No matter how small or large, this is true. 5-to-the-0-power, nine-million-to-the-zero-power, 3/5-to-the-zero-power...you name it, and it's equal to 1. Why should this be so?

Consider positive exponents: 5-squared (five-to-the-second-power or 5x5) is equal to 25 because (we are told by our teachers) the square (exponent of two) means that you have two factors of the number 5 (i.e. you multiply 5 times itself). Thus, 5-to-the-fourth-power would mean 5x5x5x5 (or 625).

Negative exponents are a little trickier: 5-to-the-negative-second-power is equal to 1/25 because a negative power is (supposedly) defined as the exponent of a number that is the denominator of a fraction with a numerator of 1. In this case, 5-to-the-negative-second-power is equal to 1 divided by 5-squared, which is 1 divided by 25, which is 1/25.

But what possible meaning can there be to multiplying a number by itself zero times? There is no such mathematical operation, is there? And even if there were, why should the result always be 1??

Yet, when we are taught to multiply by adding exponents, we have to do it that way so that the math will come out right. For instance: 6-squared times 6-squared equals 6-to-the-fourth by the exponent rule, and (6x6)x(6x6) = 36x36 = 1296, which is 6-to-the-fourth. No problem there. And: 6-to-the-zero times 6-squared equals 6-squared by the exponent rule, and 1x(6x6) = 1x36 = 36, which is 6-squared.

So, what is the deal with zero exponents? Why do they work? Are they just a "convenient fiction"--something to help the math "come out right"? Or is there something that our math teachers aren't telling us?

The confusion comes in the way that exponents are described. The basic function of an exponent is to relate the operations of multiplication and division not (primarily) to the number of which it is an exponent, but to the unit number, 1. (This, of course, is the basis of the much-dreaded "scientific notation," which I think is how students should be taught exponents, rather than the more colloquial "times-itself" approach.)

This relation to the unit 1 is explicit for negative exponents: x-to-the-minus-3rd-power, for example, is 1 divided by x-to-the-3rd-power. For positive exponents, it is left implicit (since 1 is the multiplicative identity), but is more fully stated, for example, x-to-the-3rd-power is 1 times x-to-the-3rd-power. (x-to-the-3rd = 1 times x-to-the-3rd) .

In light of this, a zero exponent's non-fictional relation to the unit 1 becomes clearly plausible. Here's how it works:

For any real number, r, a positive exponent, n, indicates that the unit 1 is to be considered as having been multiplied by r a total of n times. A negative exponent, -n, indicates that the unit 1 is considered as having been divided by r a total of n times.

A zero exponent, by contrast, indicates that the unit 1 is considered out of any multiplicative or divisive relationship to the base number r. We are to consider only the unit 1 and to refrain from combining it (by multiplication or division) with the exponent's base number.

Now we can see the root of the confusion inherent in the way most of us are taught about zero exponents. The unit 1 should be considered not as being multiplied (or divided) by some base number "zero times," but as itself, i.e., as not having been multiplied (or divided) by the base number any times.

I apologize to the mathematicians in the group for the inelegance of this explanation. I hope it is clear enough for everyone else.

===============================================================

Comments are welcome.

Interestingly, several years ago, Leonard Peikoff announced in a QA session that one of his "brilliant" students had reached a discovery/insight similar to the one I gave in 1995 and posted on my website shortly after. I guess it was just "in the air" or something...

REB

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## Gregory Browne

But that is not a foundation for counting: we understood how to count long before we had set theory.

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## Gregory Browne

Modern Predicate Logic interprets universal generalizations as conditionals (which is good when dealing with kinds), and conditionals are part of Propositionsl Logic, and modern Propositional Logic, in Russell's version, mishandles the conditional, interpreting it is a material conditional, and so that affects Predicate Logic.

I'm not sure what you last sentence means. 'Alethic' does literally mean having to do with truth, but unfortunately 'alethic modalities' is used to refer to necessity, contingency, possibility, impossibility, actuality, etc.

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## merjet

Apology accepted. It's really quite simple. Let x and y be any real numbers, x not 0. (x^y)/(x^y) = x^(y-y) = x^0. Clearly (x^y)/(x^y)= 1. Hence, x^0 = 1.

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## BaalChatzaf

The essence of counting is setting up a one to one correspondence between two sets. One set is the set of integers between 1 and n for some n. The other set is, say for example, the set of cows in your pasture. That is how you count the cows.

In days of yore we counted by matching a pile of pebbles to whatever it is we needed to count. Or we used our fingers for small numbers.

Representing large numbers by means of a numerical notation happened much later. The first attempts at that was to commandeer the alphabet. Later on we used the abacus and then positional notation.

Ba'al Chatzaf

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## Michael Stuart Kelly

Merlin,

I had to look up lattices to understand this and things started spinning. I am curious about something. I read that the symbol ^ means "and" between true symbols for the group to be true. So if I take your equation and spell it out in words, but let x=6 and y=7 (and presuming that both 6 and 7 are true), here is what I come up with:

The true group of 6 and 7 divided by the true group of 6 and 7 equals the true group of 6 and the subgroup of 7 minus 7, and this equals the true group of 6 and 0. Clearly the true group of 6 and 7 divided by the true group of 6 and 7 equals 1. Hence, the true group of 6 and 0 equals 1.

Hmmmm...

Something didn't work for me.

Michael

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## BaalChatzaf

See http://en.wikipedia.org/wiki/Modal_logic

There are other modalities besides truth, falsity, possibility and necessity. There are the epistemological modalities such as known/unknown and temporal modalities such as sometime, always, eventually and never. Then there are deontic modalities such as permitted/forbidden/optional. There are modalities such as likely, probably. There are is believable/unbelievable, the modalities relating to plausibility of propositions.

Unfortunately once you get beyond the the modalities of true/false the expressions are no longer easily analyzable in terms of their components. In short, no truth tables and such like contrivances.

There are other extensions of logic. One interesting extension has to do with -entailment- where there has to be a semantic connection between the premises and the conclusion. With material implication we get non-sense such as "the moon is made of green cheese" implies "George Bush is the president". Unfortunately, under material implication a false statement implies any statement. Entailment addresses the semantic content of the components of a conditional. See http://en.wikipedia.org/wiki/Entailment and the references therefrom.

Ba'al Chatzaf

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## merjet

No, ^ meant exponentiation or "to the power of", as in some computer languages. Context!

Edited by Merlin Jetton## Link to comment

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## tjohnson

Here is another 'intuitive' way of looking at it, take a number like 123.45

In positional notation (base 10) this number breaks down to this.

1x10^2 + 2x10^1 + 3x10^0 + 4x10^(-1) + 5x10^(-2)

So you see it's right there in front of us, 3x10^0= 3 therefore 10^0 =1

Of course this doesn't prove x^0=1 for all x but it does give one a good idea about why it must be so, I think.

Edited by general semanticist## Link to comment

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## Michael Stuart Kelly

Merlin,

I don't program. Sorry.

But with your explanation, I was able to do the math easily in my mind.

Cool.

(Cool for Roger's explanation, too.)

Michael

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