Science a GoGo's Home Page Forums General Discussion Physics Forum Hitch-hikers' Guide to Infinity

BTW; how many examples can you find of scientific concepts for which people were burned at the stake that didn't turn out to be right.

Well now. Bill S. is a duck. Bill S. is wood. Therefore Bill S. is decoy duck. Wooden decoys can be valuable. We need to see how much we can sell him for on EBay.

Bill Gill

Therefore Bill S. could be valuable; best not burn him at the stake.

I have been busy for a few days, and so did not respond to your posting. I do, in fact, agree with all the above statements that I have quoted from you.

IMO, there is nothing unscientific in your assumptions. The 'infinite' part may be left to the philosophers to speculate, and the finite part, to the scientists to explain logically.

Adding a little more (based on my theory): The infinite space and the infinite time remain as the arena. A 'very large number' of universes exist in the infinite space. Taken as a whole, the 'Ensemble', containing 'the billions of universes' does not change with time. Each universe is a single grain of matter; it is finite in space and remains pulsating (has a finite period of pulsation). During pulsation, changes happen within the universe, but it always remains as a 'system of masses'. During expansion, masses change from hot to cold (I propose cold energy), and during contraction, the masses change from cold to hot.

I like the first sentence.

I have a few problems with the second bit. If they are not addressed in "Part 5", we can come back to them.

I said I would try to keep Part 5 short - some hopes - but with any luck it will not be infinite!

Part 5

We have to consider that if we divide or multiply infinity by anything, or if we add anything to it, or subtract anything from it, it still remains infinite. This line of reasoning may seem to militate against anything I have said about the division of infinity leading to the production of finite parts. However, all it really does is establish that infinity cannot be subjected to mathematical processes. Of course, it can be argued that mathematical infinities are quite at home in mathematical processes, but I would contend that this just serves to underline the profound difference between mathematical infinities and physical infinity.

We need to look at the idea of performing any mathematical functions on a physical infinity. If it is truly infinite it is the sum total of all that exists. We cannot add anything to it, because there is nothing to add. We cannot subtract anything from it, because there is nowhere outside it that we could take anything. To talk of multiplying or dividing such an infinity is simply a conceptual exercise that has no real meaning.

Another question presents itself here: Outside of mathematics, does it make sense to talk of an infinity of any specific thing? Elsewhere I have reasoned that it makes no sense to talk of an infinite number of finite objects. What about infinite space, infinite time, infinite matter or infinite energy? It might seem reasonable to argue that it is not possible to have any one without all the others. For example, it would not be possible to have infinite matter without infinite energy (because E=mc^2), then you would need infinite space in which to accommodate it, and infinite time in which it would have to exist, since it must always have been. Perhaps Galileo was right, perhaps our finite minds are poorly equipped to grasp the concepts of the infinite. However, this should not stop us from trying.

Because Hilbert’s Hotel is a firm favourite with those who seek to gain a hold on infinity, or just to perform mathematical “tricks” with infinity, it needs a brief mention. Hilbert’s Hotel is a hypothetical hotel with infinitely many rooms, all of which are occupied; obviously by infinitely many guests. Another guest arrives looking for a room. A typical example of the way in which this guest is accommodated is found at: http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel.

“Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.”

This is little more than a verbal “conjuring trick”, if n = infinity, Hilbert’s trick works because we never have to consider what happens to the nth guest who is moved out of the nth room. Obviously there cannot be an nth room containing an nth guest, because we can never count to infinity, nor can there be a room beyond the nth room, so the whole thing is nonsense. If there were a last room into which the last guest could be moved, the new guest could be put in that room and the infinite shuffling would be totally unnecessary. It might be argued that this is a thought experiment, so it does not need to be practically possible. Even if one accepts Terry Pratchett’s (1989) definition of a thought experiment, that it is: “One that you can’t do, and which won’t work”, it has to bear some relevance to the real world. Hilbert’s Hotel falls far short of that. By what lunacy would anyone suggest that an infinite hotel could fit into a finite universe? What is an infinite number of people? If you already had an infinite number of people in the hotel, where would another one come from? (Forget mathematics; think about a real situation).
(BTW, Pratchett’s explanation, in this same book, of the “Schrödinger's cat” thought experiment is a “must read” for anyone with an interest in quantum physics, and a sense of humour).

The concept of infinity could never be kept permanently out of the realms of mathematics, and rightly so. I often think it is a shame that they didn’t change its name when they turned it into something that could be used in calculations. As we saw when considering renormalisation, anything other than a “doctored” infinity makes nonsense of mathematical calculations.

Two of the eminent mathematicians who, before Cantor, attempted to tame the infinite were Galileo and Leibniz. A good comparison of their respective approaches can be found at:
(http://steiner.math.nthu.edu.tw/disk5/js/history/infinity.pdf)
so there is no need for extensive repetition here. However, a few relevant extracts may help to clarify certain points. Galileo was quite clear that one could not progress from finite to infinite. “The transition cannot be realized step by step by continued divisions of the divisible, because such successive divisions do not lead to a last division.”

Regarding the nature of this transition Galileo held that: “When a terminated quantity proceeds to the infinite, ‘it meets with an infinite difference, what is more, with a vast alteration and change of character’. And indeed: The corresponding notions result from each other by logical negations: finite becomes infinite, divisible becomes indivisible, quanta become non-quanta”.

Significantly, Leibniz considered “the identification of an infinite number – on condition that there is such a number – with zero, and not, as Galileo had done, with unity”. Identification of infinity with unity would “assume the existence of an actually given infinite number”, which would be absurd.

A major advance arising out of Leibniz’s work was that he facilitated inclusion of indivisibles in mathematics. The way he did that is illustrated in his definition of straight lines as rectangles for the purpose of calculations of space. He lays “….the foundations of the method of indivisibles in the soundest way possible, and it supplies the proof of the method of indivisibles which enables us, ….. to find the areas of spaces by means of ‘sums of lines’. He explained these sums as sums of rectangles with equal breadths of indefinite smallness”. Obviously this is a sleight of hand trick which, while it works perfectly well in its appropriate context, did nothing to change Leibniz’s “lifelong rejection of the actual infinite in mathematics”.

A significant passage, relating to Leibniz’s work, from the article linked above is the following. “The demonstrations of the forty- five theorems that follow are based on this concept of infinitely small and infinite quantities, that is, quantities which are, according to his definition, positive, but smaller than any given quantity or larger than any given quantity. The decisive aspect is that they are quantities, fictive ones certainly, because they were introduced by a fiction, but quantities nevertheless. It does not matter whether they appear in nature or not, because they allow abbreviations for speaking, for thinking, for discovering, and for proving.”

It is not my intention to attempt to diminish the importance of Leibniz’s work. I have neither the mathematical skill, nor the wish to do that. I seek simply to show that infinitely small and infinitely large “quantities” are mathematical tools, and that Leibniz was well aware of that fact. His own assertion that: “We need not take here the infinite in the strict sense of the word” is probably all the testimony we need in that regard.

The way in which circuitous arguments can develop when trying to merge infinity and mathematics is illustrated in this quote from the linked article: “Hence a contradiction with the Euclidean axiom on the part and on the whole seems to emerge, because a part equals the whole. This result, however, was absurd in Leibniz’s opinion; it came about only because the consideration proceeds from a last abscissa which does not exist in reality, or in other words, because the indivisible, the point, is identified with an infinitely small quantity, even though both notions differ basically from each other”.

One thing emerges with certainty; infinity “in the strict sense of the word”, cannot be manipulated mathematically. This leads to a startling realisation. No part of infinity can actually be distinguished from any other. This is tantamount to saying that no part is different from any other. In other words; every part of infinity – if there could be said to be parts – would be the entirety.

The part must be the whole.


1989. Pratchett Terry & Jolliffe Gray. The Unadulterated Cat.

Sounds like what I have in mind when I speak of immanent and transcendent Being, G.O.D.

=====================================
Earlier, Orac said
=============================================
THE BEGINNING OF INFINITY, by David Deutsch
=========================
http://www.brainpickings.org/index.php/2011/07/27/the-beginning-of-infinity-david-deutsch/
____________________________________
Like DD, I do not believe in a god. I simple say that I know and experience that G.O.D. is. My role, as a bit of infinity, is to willingly choose to tune into and connect with the entirety. BTW, "connect with" is the meaning behind the Aramaic (common language in Jesus' day) word, slaha, "to pray".

Yes I know and have read the original book by David Deutsch the fabric of reality.

I did get a laugh out of one of the book reviews from Peter Forbes of the independant.



What hawkings said was “The threat of the Earth being hit by an asteroid is increasingly being accepted as the single greatest natural disaster hazard faced by humanity”

He is a phsicist he is not stupid enough to think you can blow an asteroid up with a nuclear weapon.

To explain Rev there is no atmosphere on an asteroid most likely, therefore no big shockwave and huge bang just a bright flash and lots of radiation which would look remarkably similar to a small sun :-)

Only in the movies do you blow asteroids up with nuclear weapons!

Even if you could blow up an astreoid, by the time it was close enough for you to reach it, wouldn't the resulting shower of bits be as destructive as the whole asteroid?

I do not agree with the term 'cannot be manipulated mathematically'. The relation between zero and infinity is aesthetically beautiful, and mathematically logical. The whole of mathematics is conceptual and not real. The concepts of negative numbers, imaginary numbers, etc. are aesthetically as beautiful as the concept of infinity. Actually, zero is not the other end of infinity, it is 'negative infinity' that lies at the other end. Mathematically, infinity does not represent the whole, it represents the concept of limitlessness. Mathematically, part cannot be the whole, we can take parts from the whole, and the whole gets reduced by that much.

However, I think that 'infinity cannot be manipulated physically'. Physically, infinity represents the whole. So we can leave the infinity, as I have suggested earlier, to the philosophers to speculate (because it cannot be manipulated logically). Physically, the part has a representative nature of the whole; and explaining the part tantamount to the explaining the whole, and that is what is expected from the scientists. The concept that 'the part is the whole' comes under philosophy; we can say either 'yes' or 'no' and argue, and never arrive at any logical conclusion; in the end we will have to accept that it is just a 'matter of belief'.

Finiter, to some extent we seem to be saying the same thing here, but arriving at different conclusions.

You say that “Mathematically, infinity does not represent the whole, it represents the concept of limitlessness”. This is precisely what I have been arguing on this forum, and in other places, for a long time. Such an “infinity” can, of course be manipulated mathematically, it is just limitless, a mathematical “infinity”.

You say that “The whole of mathematics is conceptual and not real.” This is true, but mathematics has been described as the language of nature. As a vehicle “for speaking, for thinking, for discovering, and for proving” it is unsurpassed. Mathematics can be applied to the 4-dimensional (or 3-D if you prefer) reality we observe, and can be extrapolated to the multi-dimensional “realities” we can imagine. Without some sort of renormalisation, or replacement by the concept of limitlessness, though, infinity is closed to mathematics, and, as you say, to physical manipulation.

I completely agree with you that “zero is not the other end of infinity”. In fact, I don’t believe that infinity is at one end of any sort of continuum.

Negative infinity? I think that must be a concept for the philosophers. Infinity, on the other hand, needs to be understood if we are to have any real grasp of the origin and nature of the Universe.

BS,you regard yourself as a hitchhiker, and I am just a layman interested in physics. We agree on many things, and that is very interesting.

You said, "Mathematics is the language of nature".

I think you have slightly exaggerated the role. Language is used to describe. So you are suggesting that mathematics can describe the physical world. I would suggest a lesser role; it is just a tool to extract results. The physical world should be explained physically, and for that verbal explanation will be enough. However, all physical laws should be mathematically valid. Thus, mathematics has a role in proving whether the physical model is correct or not. The role of mathematics as a vehicle for 'speaking, thinking and discovering' is minimal. The line of thought should be like this: visualize a physical model, then verify whether it is mathematically feasible; if not feasible try another physical model and so on. It should not be in the reverse order: that is, you should not impose a mathematical model (even if it agrees with the observations) on the physical world. The reason is that all mathematically valid explanations are not valid physically, whereas all physical models should be mathematically valid.

So there is a need to distinguish between mathematical and physical infinities as suggested by you. I agree that infinity is not an end. However, zero may be an end or a beginning. Negative infinity is a mathematical concept. It can have some role in philosophy. I think philosophy is linked to other branches like this: philosophy - mathematics - science - philosophy, thus forming a cycle. Mathematics is conceptual, science is real, and philosophy, both conceptual and real.

You said, "Infinity, on the other hand, needs to be understood if we are to have any real grasp of the origin and nature of the Universe."

As you said initially, 'something cannot come out of nothing'. So there is no origin; or, origin will always remain a mystery. In explaining the nature of the universe, infinity has no real role. That is, if a physical theory leads to infinity (other than the basic assumptions that space and time are infinite, and for which there are no explanations), the theory is wrong; there should not be any re-normalizations.

Finiter, I think we could agree on more points than may be immediately obvious.

For example: I said that mathematics has been described as the language of nature, not that I thought it was the language of nature. I agree that words can do a very good job where actual description is concerned, but as one who has greater facility with words than with figures, I suppose I would.

You say: “…all mathematically valid explanations are not valid physically”. I agree absolutely, and think there are very few areas in which this is more obviously true than in dealing with infinity.

Where we do seem to differ is in our respective attitudes to the role of infinity in explaining the origin of the Universe. We can say the “origin will always remain a mystery”, and that is a very reasonable position to take. On the other hand, we can explore the whole question to see if there is a logical and scientific link between the infinite and the finite. Such an endeavour might involve making some basic assumptions, but as long as one remembers that they are just assumptions and tries to verify them scientifically, that seems a legitimate way to go.

“…there should not be any re-normalizations.”

I think renormalisation is a reasonable mathematical tool in that it seems to produce results that agree well with observation. Ancient peoples managed to fit supernatural explanations to physical observations, such explanations remained “valid” until better, more scientific, explanations were found, often these explanations were superseded. Possible, renormalisation will be of value until something better is discovered. I would say, keep it, use it, but don’t “deify” it.

In the movies the small bits burn up harmlessly in the atmosphere .... I mean those nuclear bombs are powerful apparently they blow the meteor to bits apparently :-)

But yes Bill S I will leave the offical calculations to NASA and they put the limit at 400 meters. Anything bigger than 400m could probably not be nullified by any amount of nuclear weapons at best you would probably break it into big pieces and those are still going to smash into earth.

2005 YU55 which whistled past this month was of this sort of size.

Thread drift is a wondrous thing! How did we get from infinity to blowing up asteroids in the movies?

Someone must have some strong objections to the infinite stuff. Where is Kallog when you need him?

I think our our differences are narrowing further. Is there a logical and scientific link between the infinite and the finite?

Here, we have to make clear the meaning of 'assumption'. Without going into the dictionary-defined meaning, I would say that an 'assumption' has no proof; ie, it cannot be logically proved. And, a logical proof, in the context of physical world, is a mathematical proof. So an 'assumption' has no mathematical proof and so any assumption is not part of physics in the true sense.

However, we can have 'logical assumptions'. We can use a 'logical assumption' as a starting point when we propose a theory. But, before concluding, we will have to logically prove the assumption. So a 'logical assumption' is provisional, and we are liable to prove it before we say 'it is finished'.

Now, coming to the point, your statement 'Such an endeavour might involve making some basic assumptions ........ a legitimate way to go' is right. You are suggesting that we have to make a 'logical assumption', and later try to prove it. However, in my opinion, only assumptions (which cannot be logically proved) are possible; I am ruling out the possibility of a 'logical assumption' that can link infinity with finiteness. I think this is the main difference between our views.

Re-normalization is a mathematical tool. Though the process is complex, the logic behind it is simple: it can extract results. As such, it is useful and I am not at all questioning its validity. I am only saying that to explain the basics of this physical world, neither 'the probability equations' nor 'the re-normalizations' are required. The reason is that the physical world we confront (ie, our universe) is deterministic and finite.

Based on my theory, the 'Finiteness theory' (sorry, I have to refer to my theory just to make it clear that my argument is based on an alternate theory), mass, length and time (MLT) are the only basic parameters. To explain the fundamentals of this physical world, we require only these parameters. These parameters are deterministic and do not lead to any infinities. However, the permutations/ combinations of these parameters and their squares,cubes, etc., will create a very complex state (like that of a living being like us interacting with its surroundings)that we have to use 'probability equations' and 're-normalizations' to extract results. But basically the physical world is deterministic, and that is why the probability equations and the re-normalizations give requied results.

Finiter, I like your distinction between "assumptions" and "logial assumptions".

As far as the possibility of a logical assumption that can link infinity with finiteness is concerned; I suspect there should be a logical link somewhere, but it obviously needs more thought.

Just come back from walking dog. A thought struck me while out. A long time ago I reached the conclusion that nothing finite could became infinite. The obvious corollary, that nothing infinite could become finite, had somehow not registered. There has to be more to be said about that.

IMO, your line of thinking is correct. I will give examples to clarify my view. Suppose matter has a fundamental particle having finite mass and finite volume (matter is grainy), then the universe made up of matter will never be infinite, it will always be finite.

Now taking the corollary: Suppose the space is infinite, then you can never have a finite space. That is, the space cannot be grainy or there is no quantized space. From zero space(a dimensionless point) to infinity (having no limit) the space is continuous. That is, if starting from a point you increase the space by making it an expanding sphere (space is three dimensional and so we have to expand it in all directions) the space enclosed can take any value (not restricted to a subset of values). However the finiteness of each value is just conceptual, not real.

Time is also like space; it is not grainy, but it is only one dimensional. So expanding it from a point of zero time we get a line extending in both directions infinitely. For any point on the line we get a finite value, which is conceptual, but not real.

If there is only space and time and no matter, then it represents a physical 'nothingness', but it is real, consisting of infinite real space and infinite real time. Here, it may appear that physical infinity and physical zero becomes equal. Therefore, do you think that we can conclude that physically infinity and zero are equal? I have never thought of such a possibility.

However, that may be philosophical statement, implying that the real physical world is finite. Such a philosophical statement, I think, does not rule out the possibility of more than one finite physical worlds.

Can that be a logical statement (something that can be proved mathematically) ? I have to think about it.

I toyed with the idea of infinite = zero, which progressed to infinite = zero = the present.
Any thoughts?