How does one find True Knowledge?

[Sculptor1]

Sculptor1, regarding empirical knowledge, don't you think that scientific knowledge accumulates like as Hegel described thesis, antithesis, synthesis? the newer paradigm would not have been possible without the older paradigms? Isn't this a benefit of culture which is transmitted down the generations including when cultures were oral ones?

Yes - though that sounds suspiciously like progress.
But I think maybe science is the one area of human activity were progression is inherent in the process.
It is, though, wise to always acknowledge that progress in understanding and knowledge does not always lead to progress in the cultural and social outside science.

It can be argued that Homeric epic was the result of a long tradition of oral transmission. What started as an account of a war and the journey home of one of the heroes, it transformed into a fantastic story with talking horses, witches, battling gods, strange creatures etc..
Oral herb lore was no different.
Oral transmission is poor without verification. And scientific verification is hard fought for.
Let's say that parent tells child about a herb that helps heal wounds. Next generation that child uses the herb but for other reasons the patient dies. Child no longer uses that herb. Knowledge lost.
Or parent uses herb B ,but patient survives for another reason that herb B. Parent continues to use useless herb B and tells child to do the same. Transmission of bad knowledge.
Without recording on paper or other media; without double blind testing, formal experiments all recorded, each generation has to learn from its own experience, which can be faulty.

When it comes to skills such as iron forging where the practical results are plain to see, oral transmission can be better than written, since there are some techniques you just have to do with your hands.

[devans99]

Well spacetime was not there 14 billion years ago according to the Big Bang theory…

"If space is infinite and you shrink all distances by a factor of two, the size of space becomes half of infinity, and that is still infinite. So although everything gets closer together and the densities get ever higher as you head further back in time, the overall size of the universe stays infinite; things get dense everywhere on an infinite spatial expanse. This yields a rather different image of the big bang.
Normally, we imagine the universe began as a dot…in which there is no exterior space or time. Then, from some kind of eruption, space and time unfurled from their compressed form and the expanding universe took flight. But if the universe is spatially infinite, there was already an infinite spatial expanse at the moment of the big bang. At this initial moment, the energy density soured and an incomparably large temperature was reached, but these extreme conditions existed everywhere, not just at one single point. In this setting, the big bang did not take place at one point; instead, the big bang eruption took place everywhere on the infinite expanse. Comparing this to the conventional single-dot beginning, it is as though there were many big bangs, one at each point on the infinite spatial expanse. After the bang, space swelled, but its overall size didn't increase since something already infinite can't get any bigger. What did increase are the separations between objects like galaxies (once they formed)."

(Greene, Brian. The Fabric of the Cosmos: Space, Time, and the Texture of Reality. London: Penguin, 2005. pp. 248-9)

The maths of infinity just does not hold together to support the above view:

∞ / 2 = ∞ implies 'the whole is greater than the parts' is false - which is clearly nonsense.

'After the bang, space swelled, but its overall size didn't increase since something already infinite can't get any bigger.' - he clearly contradicts himself with the claim that space swelled (=increased in size) but did not get any bigger. ∞ * 2 = ∞ makes no sense either - in english it says 'there exists something that when you change it, it is not changed'. Such a belief is incompatible with reason.

'something already infinite can't get any bigger' - I agree with that statement. The universe is expanding hence the only reasonable conclusion is that it is not infinite.

BTW, contrary to what Cantor said, there are not multiple, different sized infinities. Actual infinity must be defined as a quantity larger than any other quantity (if something so spiritual can claim to have a logical definition). That clearly precludes two different infinities - they would have to be larger than each other - a contradiction.

Whether spacetime started as a dot or a larger but finite compressed region is not clear to me. I think perhaps a dot of finite density (since there must be a finite amount of energy/matter in the universe - we have established that space is finite so that rules out infinite energy/matter because it would lead to infinite density everywhere).

[Bluemist]

"If space is infinite and you shrink all distances by a factor of two, the size of space becomes half of infinity, and that is still infinite. So although everything gets closer together and the densities get ever higher as you head further back in time, the overall size of the universe stays infinite; things get dense everywhere on an infinite spatial expanse. This yields a rather different image of the big bang.
Normally, we imagine the universe began as a dot…in which there is no exterior space or time. Then, from some kind of eruption, space and time unfurled from their compressed form and the expanding universe took flight. But if the universe is spatially infinite, there was already an infinite spatial expanse at the moment of the big bang. At this initial moment, the energy density soured and an incomparably large temperature was reached, but these extreme conditions existed everywhere, not just at one single point. In this setting, the big bang did not take place at one point; instead, the big bang eruption took place everywhere on the infinite expanse. Comparing this to the conventional single-dot beginning, it is as though there were many big bangs, one at each point on the infinite spatial expanse. After the bang, space swelled, but its overall size didn't increase since something already infinite can't get any bigger. What did increase are the separations between objects like galaxies (once they formed)."

The maths of infinity just does not hold together to support the above view:
∞ / 2 = ∞ implies 'the whole is greater than the parts' is false - which is clearly nonsense.

'After the bang, space swelled, but its overall size didn't increase since something already infinite can't get any bigger.' - he clearly contradicts himself with the claim that space swelled (=increased in size) but did not get any bigger. ∞ * 2 = ∞ makes no sense either - in english it says 'there exists something that when you change it, it is not changed'. Such a belief is incompatible with reason.

'something already infinite can't get any bigger' - I agree with that statement. The universe is expanding hence the only reasonable conclusion is that it is not infinite.

BTW, contrary to what Cantor said, there are not multiple, different sized infinities. Actual infinity must be defined as a quantity larger than any other quantity (if something so spiritual can claim to have a logical definition). That clearly precludes two different infinities - they would have to be larger than each other - a contradiction.

Whether spacetime started as a dot or a larger but finite compressed region is not clear to me. I think perhaps a dot of finite density (since there must be a finite amount of energy/matter in the universe - we have established that space is finite so that rules out infinite energy/matter because it would lead to infinite density everywhere).

Physics is purely mathematical and cannot be precisely translated or explained in any ordinary language. What makes Brian Greene's book valuable is that it attempts to present the flavor of physical (mathematical) thought to readers of ordinary language. It is not the case that his words, even if these were completely understood, and I claim no such thing for myself, would present an equivalent representation to the real thing.

Also, note that Greene's description attempts to shift the perspective of the Big Bang from an external point in some god's x-dimensional space-time to one looking around from the inside. This is the only real physical view of the actual world.

A point is infinitely small whereas the view from inside the 'point' is infinitely big. To naïve native human thought, neither can be understood nor rightly described because they are not part of our ordinary experience. Nevertheless, mathematics far surpasses experience and imagination into remote abstractions built on top of less remote ones.

Infinities can't come in ordinary sizes and comparatives such as bigger and smaller or alike and different. To agree or disagree with Cantor, one would have to do it in the language of his math.

[RJG]

Are there an infinite number of points on a number line from 0 to 1?

Are there an infinite number of points on a number line from 0 to 2?

Which infinite number is bigger?

[devans99]

Are there an infinite number of points on a number line from 0 to 1?

Are there an infinite number of points on a number line from 0 to 2?

Which infinite number is bigger?

A point is length zero so we have:

1 / 0 = UNDEFINED
2 / 0 = UNDEFINED

Actual infinity does not exist.

[RJG]

Actual infinity does not exist.

If so, then…
...what is the last number at the end of the number line?
...how many times can a segment be divided into smaller segments until there are no more smaller segments?

Are there an infinite number of points on a number line from 0 to 1?

Are there an infinite number of points on a number line from 0 to 2?

Which infinite number is bigger?

A point is length zero so we have:

1 / 0 = UNDEFINED
2 / 0 = UNDEFINED

This reply is a non-sequitur. I asked about the number of points, not the "length" of points.

[RJG]

New question:

Can there be an infinite number of segments on a number line from 0 to 1?

Can there be an infinite number of segments on a number line from 0 to 2?

If so, then is one infinite number two times bigger than the other?

[devans99]

New question:

Can there be an infinite number of segments on a number line from 0 to 1?

Can there be an infinite number of segments on a number line from 0 to 2?

If so, then is one infinite number two times bigger than the other?

I can think of two possible answers, depending on the definition of 'point' used:

There number of possible segments between n points is n × ( n -1 ). We have established above that n must be finite - a point must have a non-zero size else there are an undefined number of points on a line segment - so n × ( n -1 ) must be finite too.

Or if we stick with the (impossible) length zero point, then the nature of a continuum is that a larger part of a continuum must be identical in structure to a smaller part of the continuum. In essence, a true continuum would flaunt 'the whole is greater than the parts'. So both the larger and smaller segment would have an equal infinity of sub-segments. But of course this requires the impossible definition of a point and flaunts a well trusted axiom.

[devans99]

Whoops I meant n x (n-1) / 2

[RJG]

Can there be an infinite number of segments on a number line from 0 to 1?

Can there be an infinite number of segments on a number line from 0 to 2?

If so, then is one infinite number two times bigger than the other?

I can think of two possible answers, depending on the definition of 'point' used:

Please note that "points" has nothing to do with this question. I specified "segments".

For example, on a ruler there are:
-- 8 (1/8") segments between the 0 and 1" mark, and
-- 16 of these same (1/8") segments between the 0 and 2" mark.

16 is twice as big as 8. Does this relationship (2X) continue on through into infinity?

In other words, if we slice each of these segments in half (to 1/16" each), the 2X relationship STILL remains! If we continue dividing/splicing the segments infinitely, doesn't the 2X relationship still remain? And if so, doesn't this mean that "infinite" numbers can have different values?

[devans99]

Please note that "points" has nothing to do with this question. I specified "segments".

?

A segment is defined by the two points at either end... so you can't consider segments without considering points.

[RJG]

Devans, I'm not asking about the number of non-dimensional "points", I'm specifically asking about dimensional "segments". See above for my added clarification.

[devans99]

Please note that "points" has nothing to do with this question. I specified "segments".

For example, on a ruler there are:
-- 8 (1/8") segments between the 0 and 1" mark, and
-- 16 of these same (1/8") segments between the 0 and 2" mark.

16 is twice as big as 8. Does this relationship (2X) continue on through into infinity?

In other words, if we slice each of these segments in half (to 1/16" each), the 2X relationship STILL remains! If we continue dividing/splicing the segments infinitely, doesn't the 2X relationship still remain? And if so, doesn't this mean that "infinite" numbers can have different values?

But equally, we could have marked off 16 smaller segments on the 1" ruler and 16 on the 2" ruler and then continued slicing to each to infinity... The only way the 2" ruler could ultimately have more segments than the 1" ruler is if the rulers are made of discrete material (which they are in reality), but then both rulers would have a finite number of segments. Otherwise we can keep dividing both infinity so I don't think that it can be said they are different sizes of infinity (Cantor, for what its worth, has aleph-one as the 'size' of all continua - whatever their size).

Above forms an argument for discrete space - logically the 'information content' of 1 light year of space should be larger than the information content of 1 centimetre of space (whole is greater than the parts). But with a continuum, both can be subdivided to an equal infinity suggesting both have the same information content... which seems very wrong.

[Bluemist]

It seems that this problem is not new:

And with this [Cantor's Set Theory] I believe to define something related to the Platonic eidos or idea, as well as that which Plato in his dialogue Philebus or the Supreme Good" calls mikton. He contraposits this against the apeiron, i.e., the unlimited, indeterminate which I call inauthentic-infinite, as well as against the peras, i.e., the limit, and he declares the [mikton] an orderly "mixture" of the latter two.

As Aristotle was wont to narrate, this was what happened to the majority of the people who heard Plato’s lecture On the Good. Each came expecting to learn something about the things which are generally considered good for men, such as wealth, good health, physical strength,
and altogether a kind of wonderful happiness.

But when the mathematical demonstrations came, including numbers, geometrical figures and astronomy, and finally the statement. Good is One it all seemed to them, I imagine, utterly unexpected and strange; hence some belittled the matter, while others rejected it.

[RJG]

But equally, we could have marked off 16 smaller segments on the 1" ruler and 16 on the 2" ruler and then continued slicing to each to infinity…

But then you would no longer be comparing apples-to-apples. If the segments in each group are not the same, then you are comparing apples-to-oranges, which is an invalid comparison.