Does infinity exist in the physical world?

Upisoft said:

That's interesting. Do you learn history of matehmatics in your regular course? We have it as separate course, that I didn't pick. But now I see it could be interesting to know what errors they made in the past.

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Close. I teach History of Mathematics as part of a wider course - I'm a lecturer in Mathematics Education. And yes, I do think it's illuminating to see that maths is not some immutable and timeless discipline but a developing and growing field of knowledge.

Upisoft said:

That's interesting. Do you learn history of matehmatics in your regular course? We have it as separate course, that I didn't pick. But now I see it could be interesting to know what errors they made in the past.

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To answer your question, somewhere along the way, as a part of my regular calculus studies, I did learn about "infinitessimals" and how they posed a problem for the early formulations of calculus. It was part of the discussion about how Newton and Leibniz both developed calculus independently and simultaneously.

They posed a problem because they were used in contradictory ways. Sometimes they were considered so small they could simply be ignored, but other times they needed to be taken into account to get the proper result.

MarcusHill said:

It depends on what you mean by "infinity", "exist" and "the physical world". Sorry if that sounds like a coput, but I'm a mathematician.

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You know, I do happen to have a math degree myself (though I'm probably not nearly as knowledgable as someone like yourself), so I'm somewhat interested in this discussion. Speaking as a physicist, though, I might make a slightly stronger statement, and say that infinity does not exist in the physical world. In means much the same thing in science as it does in mathematics: it refers to a limiting process.

yasic said:

Also, one could claim that the universe has infinite length. As in it has no start or finish. We do not have any way to prove or disprove that at the moment either.

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Actually, it's commonly believed today that the universe is finite in size. There are even estimates on the mass of the universe. So it's entirely within the realms of cosmological theory that the universe is finite in size. As far as I can tell, this is in fact the most likely possibility.

Upisoft said:

You know there is no such number that is smaller than any real number. I thought you said you're mathematician...

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MarcusHill said:

That's why I said "they're not in vogue". They had to be posited and used in the early development of calculus, but even then mathematicians realised they weren't really a tenable concept. That's why there was a century of refinement in order to reformulate the foundations of calculus in terms of limits.

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Interestingly, I happen to have an old calculus textbook (=from the sixties) called "Calculus: An Infinitesimal Approach." As Marcus said, many of the early pioneers of calculus, such as Newton, Leibniz, the Bernoulli guys, etc., genuinely believed in infinitesimals. In the next century, Weirstrass formulated the rigorous "epsilon-delta" limit definition, which eliminated the need for infinitesimals, and in fact modern calculus courses do not even entertain the idea. But this textbook of mine revives the idea, and introduces the concept of infinitesimal numbers. Apparently the idea can be rigorously formulated.

Not that physicists care about rigorous formulation. All this time we've been under the impression that the familiar "dx" from calculus is a bona fide number.

arunma said:

You know, I do happen to have a math degree myself (though I'm probably not nearly as knowledgable as someone like yourself),...

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I'm always in two minds about revealing the extent of my experience and qualifications in discussions like this one. Even more than in other fields, mathematical arguments should stand by themselves without needing to appeal to authority.

... so I'm somewhat interested in this discussion. Speaking as a physicist, though, I might make a slightly stronger statement, and say that infinity does not exist in the physical world. In means much the same thing in science as it does in mathematics: it refers to a limiting process.

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Chances are that no collection of physical objects is countably finite, even the number of quarks in the universe, and no measure (distance, volume, mass...) is infinite on any given scale. However, there are still an infinite number of points on a line you can physically draw with a pencil. The question is whether these ideal mathematical points can be said to "exist" in a "physical" sense - and that is very much down to semantics.

"It possibly exist"

There are concepts that when spelt out do deal with infinity. Such as limits, a curve approaching a value.

In real life we have "infinite expansion," possibly if the Universe is flat without enough mass to bring it back on itself and with our knowledge today that could be the case. We use to think that black holes were infinite in mass and density and created a rip in space/time. Now we know that this isn't the case seeing as they only have a mass that is enough to act on space time and through the study of quantum mechanics on the event horizon, shown to evaporate over time.

I can see where the repulsive force of gravity at the beginning of the Universe could be sufficient enough to have space/time expand for infiniti given we are approaching symetry. However, when Theoretical Mathematicians and Physicist see infinity spelt out as the end result of one of their equations, they dismiss it because they have too. It means it will not be an answer we can use, and has shown to be in error in the past. For that matter, infinity should never be expressed anywhere in a fully rendered equation.

Further note:

Scientists and Mathematicians realize that infiniti expressed in an equation means there is something missing or something misunderstood. Some real life individuals and groups of individuals use this to their advantage here on Earth for that same reason.

MarcusHill said:

I'm always in two minds about revealing the extent of my experience and qualifications in discussions like this one. Even more than in other fields, mathematical arguments should stand by themselves without needing to appeal to authority.

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Oh of course. To be honest I don't know the first thing about (real) math. I prefer to leave it to you guys to talk about groups, rings, fields, and all that other stuff. I'm just stating my personal experience as my motivation for discussing what, to many, might seem like a mundane issue.

MarcusHill said:

Chances are that no collection of physical objects is countably finite, even the number of quarks in the universe, and no measure (distance, volume, mass...) is infinite on any given scale. However, there are still an infinite number of points on a line you can physically draw with a pencil. The question is whether these ideal mathematical points can be said to "exist" in a "physical" sense - and that is very much down to semantics.

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Perhaps it is just semantics. Physicists have of course estimated the number of protons (and by consequence, an approximation of the number of quarks) in the universe. It's a big number, but should we really call it infinite? I'm not sure what the mathematician's answer to this would be...

arunma said:

Perhaps it is just semantics. Physicists have of course estimated the number of protons (and by consequence, an approximation of the number of quarks) in the universe. It's a big number, but should we really call it infinite? I'm not sure what the mathematician's answer to this would be...

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I'm not a mathematican proper but if you can count the number of elements the set is not of infinite size.

Upisoft said:

Infinity is abstact concept, very useful in mathematics. Since the mathematics are some level of abstraction of the physical reality, I ask this question.

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I had a Professor that posed a question when he was trying to "explain what mathematics is". He basically said: "Mathematics is our method of relating things so that we can come to an understanding of the world. If we have two cows and two sheep we can related them by this idea of "twoness". But does "twoness" really exist?"

yasic said:

- Every real number without an infinite decimal expansion (except for zero) has 2 differently expressed numbers with infinite decimal expansions which are equivalent to the number. -

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Sounds rational.