Creationists: How do you test for "truth"?

[Polymath257]

Is infinity, as a concept, a boundary?

Not usually.

Is a "set" of infinity that has an unknowable value?

No. For example, the set of all natural numbers, {0,1,2,3,....} is an infinite set (in other words, it has an infinite number of objects). In general, sets do not have values (except, by a stretch, themselves). The results of operations may, depending on context

 

[Polymath257]

Then we have 'fraction value greater than 0'. And the mathematical answer would appear to be, 0+(1 infinitesimal).

Comment: unless you are doing non-standard analysis (which I doubt), there are no infinitesimals. This was an issue early on in the development of calculus and was (rightly) criticized by the philosopher Berkeley. But after Cauchy (early 1800s), such entities do not appear in standard treatments of limits or of calculus.

 

[sandy whitelinger]

And whatever your answer, unknowability is a wholly different thing to 'not presently known'.

There are degrees of unknowability. There is that which is unknowable to the human mind. There is that which is unknowable until certain steps are taken. There is that which if you don't know it at that moment in time then it's unknowable at point of time. These are interchangable.

 

[sandy whitelinger]

Not usually.


No. For example, the set of all natural numbers, {0,1,2,3,....} is an infinite set (in other words, it has an infinite number of objects). In general, sets do not have values (except, by a stretch, themselves). The results of operations may, depending on context

Ah.

 

[blü 2]

Comment: unless you are doing non-standard analysis (which I doubt), there are no infinitesimals. This was an issue early on in the development of calculus and was (rightly) criticized by the philosopher Berkeley. But after Cauchy (early 1800s), such entities do not appear in standard treatments of limits or of calculus.

My debt is indeed to Robinson here, who showed that a system of hyperreal numbers was consistent with ─ um, unhyperreal ─ mathematical principles. At the same time, while all but all maths above the rationals Q defies mapping onto reality, I admit I find the concept of (Cantorian) infinity/ies particularly irritating.

Do you have an alternative answer in mind for our friend sandy?

 

[blü 2]

There is that which is unknowable to the human mind.

This is the moment for your example of something in principle unknowable.

 

[sandy whitelinger]

This is the moment for your example of something in principle unknowable.

Knowing the unknowable.

 

[blü 2]

Knowing the unknowable.

That's not an example. That's just the assertion you keep repeating.

What's a non-trivial example of 'something unknowable to the human mind'?

 

[Polymath257]

There are degrees of unknowability. There is that which is unknowable to the human mind. There is that which is unknowable until certain steps are taken. There is that which if you don't know it at that moment in time then it's unknowable at point of time. These are interchangable.

I assume you mean that these are *not* interchangeable.

Of these, I would only label the first (unknowable to the human mind) as 'unknowable'.

Requiring certain steps to be taken before knowledge means it *is* knowable: by taking those steps. It is unknown, but knowable.

If you don't know something at one point in time, that in no way implies it to be unknowable. For example, when I was young I did not know the fundamental theorem of calculus. Now I do. That theorem is knowable, but I didn't know it at the time.

Your example of color to the blind man is another example of 'knowable, but unknown'. Even the blind man can know about color. he just can't *experience* color.

Finally, you make a good distinction between knowing about and knowing details. For example, we know about dark matter, but there is a great deal we do not know concerning it. But, to know about something, you still need evidence of its existence.

What evidence do you have that there is something inherently unknowable?

 

[Polymath257]

My debt is indeed to Robinson here, who showed that a system of hyperreal numbers was consistent with ─ um, unhyperreal ─ mathematical principles. At the same time, while all but all maths above the rationals Q defies mapping onto reality, I admit I find the concept of (Cantorian) infinity/ies particularly irritating.

The Robinson model is very interesting, but deficient in many ways. In particular, it is a first order model for the axioms of the real numbers and one of many different such (at least one for each ultrafilter on the naturals). In contrast, the 'usual' set of real numbers is a second order model and is unique. This has a number of very significant consequences for existence and uniqueness theorems.

Another aspect: to create the Robinson reals, you have to already have the ordinary reals.

Do you have an alternative answer in mind for our friend sandy?

I'm not sure what the current question is.

 

[sandy whitelinger]

That's not an example. That's just the assertion you keep repeating.

What's a non-trivial example of 'something unknowable to the human mind'?

That is the example. I know the unknowable would be an assertion

 

[sandy whitelinger]

I assume you mean that these are *not* interchangeable.

Of these, I would only label the first (unknowable to the human mind) as 'unknowable'.

They are interchangeable. If the mind changes then what was unknown before could become known. Then that unknowable needed a step to be taken to become known.

This is also an example of how one moment in time freezes the unknown as unknowable at that time otherwise it would be known.

Requiring certain steps to be taken before knowledge means it *is* knowable: by taking those steps. It is unknown, but knowable.

If you don't know something at one point in time, that in no way implies it to be unknowable. For example, when I was young I did not know the fundamental theorem of calculus. Now I do. That theorem is knowable, but I didn't know it at the time.

Things are not knowable until they are known otherwise you can make no distinctions between time. Yesterday is as good as now which is as good as tomorrow. Only by speaking of the future as a concrete fact can you say you will know something.

Your example of color to the blind man is another example of 'knowable, but unknown'. Even the blind man can know about color. he just can't *experience* color.

Knowing about and knowing are not the same thing.

IFinally, you make a good distinction between knowing about and knowing details. For example, we know about dark matter, but there is a great deal we do not know concerning it. But, to know about something, you still need evidence of its existence.

The fact that it is unknown is proof that there is no evidentiary evidence. I don't see how the unknowable can be described in empirical terms, otherwise it already would have become known.

 

[sandy whitelinger]

What's a non-trivial example of 'something unknowable to the human mind'?

You will never know instantaneous sensory awareness. By the time your aware of it it's over.

 

[Polymath257]

They are interchangeable. If the mind changes then what was unknown before could become known. Then that unknowable needed a step to be taken to become known.

This is also an example of how one moment in time freezes the unknown as unknowable at that time otherwise it would be known.

Things are not knowable until they are known otherwise you can make no distinctions between time. Yesterday is as good as now which is as good as tomorrow. Only by speaking of the future as a concrete fact can you say you will know something.

Knowing about and knowing are not the same thing.

The fact that it is unknown is proof that there is no evidentiary evidence. I don't see how the unknowable can be described in empirical terms, otherwise it already would have become known.

OK, all I can say is that you seem to use the word 'unknowable' in a very strange way.

 

[sandy whitelinger]

OK, all I can say is that you seem to use the word 'unknowable' in a very strange way.

Im a strange guy. I'm a bit of a Christian Taoist. I've learned a lot through our dialogue though. Thank you.

 

[sandy whitelinger]

OK, all I can say is that you seem to use the word 'unknowable' in a very strange way.

I'm reminded of a trip to Mammoth Cave. The tour guide was asked if there was more of the cave to discover. He answered with a a chuckle, "How am I to know? They would have to discover it before I could know that."

 

[blü 2]

You will never know instantaneous sensory awareness. By the time your aware of it it's over.

Another trivial example. The concept of instantaneous sensory awareness, like the concept of yellow, the concept of extremely distant stars, the concept of the center of the earth, is fully comprehensible, is easily known. The inability to experience something directly is entirely different from the inability to know / understand something.

 

[blü 2]

The Robinson model is very interesting, but deficient in many ways. In particular, it is a first order model for the axioms of the real numbers and one of many different such (at least one for each ultrafilter on the naturals). In contrast, the 'usual' set of real numbers is a second order model and is unique. This has a number of very significant consequences for existence and uniqueness theorems.

Another aspect: to create the Robinson reals, you have to already have the ordinary reals.

I'm not sure what the current question is.

In #284 sandy asked "For example what is the first fractal value greater than 0". Asked to clarify, sandy replied that 'fractal' was a spellcheck error for 'fraction'.

So the point of proposing 0+ (1 infinitesimal) was to provide an answer within maths to the amended question, 'What is the first fraction value greater than 0'. (As well, although as I said, Cantorian infinities irritate me, for some reason the idea of one infinitieth strikes me as funny.)

 

[Polymath257]

I'm reminded of a trip to Mammoth Cave. The tour guide was asked if there was more of the cave to discover. He answered with a a chuckle, "How am I to know? They would have to discover it before I could know that."

But is that an unknowable? Certainly not. It is simply unknown.

 

[Polymath257]

In #284 sandy asked "For example what is the first fractal value greater than 0". Asked to clarify, sandy replied that 'fractal' was a spellcheck error for 'fraction'.

There *is* no first fraction greater than 0. And that is as true in Robinson arithmetic as it is in the standard model. If x>0, then 0<x/2 <x. That's all that is required for the proof.

So the point of proposing 0+ (1 infinitesimal) was to provide an answer within maths to the amended question, 'What is the first fraction value greater than 0'. (As well, although as I said, Cantorian infinities irritate me, for some reason the idea of one infinitieth strikes me as funny.)

Two issues here:

1. If you pick any specific infinitesimal, say x. Then x/2 will be a smaller infinitesimal. So the trick doesn't work.

2. Why do Cantorian infinities irritate you? It is simply a relationship between sets (the existence of a bijection) and the realization that there are infinite sets (for example, the set of natural numbers). If you then allow the construction of the power set of any set (the set of all subsets), you inevitably get the Cantor hierarchy, although perhaps not with an identification with the ordinals.