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October 16th, 2014, 04:04 PM   #1
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Should infinity be removed from mathematics?

I studied mathematics at school and as part of my computing degree, but I never felt at ease with the concept of infinity. Recently I gave the matter more thought and I reached the conclusion that infinity is not a valid concept.

Of all the areas of mathematics that I have encountered (school-level set theory, calculus, geometry, probability and so on) I could find no valid reason for using infinity. For me, its intangible nature creates mysticism where I would prefer precision and clarity. To address this, I wrote an article suggesting how we could start to remove infinity from these areas of mathematics (see: Background » Extreme Finitism).

Infinity causes problems that do not exist without it. For example, if we accept the infinite set of all natural numbers actually exists {1, 2, 3, 4…}, what is the percentage of odd numbers in this set?

The obvious answer of 50% is problematic because it implies an even number of elements whereas infinity is supposedly neither odd nor even. Another option is to say the cardinality of the set of all odd natural numbers is exactly the same as the cardinality of all natural numbers, which is far from a satisfactory answer.

Without infinity we can simply say the percentage of odd numbers in the first n natural numbers is 100*floor[(n+1)/2)]/n where n>=1 (note that 'floor' means 'round down'). Without infinity we can say this is a complete solution for all natural numbers. Without infinity we no longer have any problems.

There is a nice bullet-point summary of the main problems with infinity (as a mathematical concept) at the end of this article: Infinity and infinite sets: the root of the problem » Extreme Finitism

I am very interested to hear other people's ideas on how to remove infinity from mathematics, or possibly why it should not be removed at all. The more opinions I can get about this subject the better. Many thanks in advance.
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October 16th, 2014, 04:21 PM   #2
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How would you write the problem $\displaystyle \lim_{x \to \infty} \frac{x^2 + 2}{x^2 + 3x - 7}$ without using infinity?

-Dan
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October 16th, 2014, 05:05 PM   #3
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Originally Posted by Karma Peny View Post
Without infinity we can simply say the percentage of odd numbers in the first n natural numbers is 100*floor[(n+1)/2)]/n where n>=1 (note that 'floor' means 'round down'). Without infinity we can say this is a complete solution for all natural numbers.
This is true whether there "infinity" is admitted or not.

Without the infinite, we can't adequately deal with the very, very small or the very, very, large. To discard the infinite leaves a hole in mathematics, and to me it's a hole where much of the most interesting stuff lies. Of course, it's interesting partly because it's a tricky concept, but that doesn't mean it should be thrown out.

Discarding things because they seem too difficult or because they don't fit with your preconceived ideas is religion, not mathematics. That's not to say that one can't do valid and even interesting mathematics without admitting the infinite. Just that you have to be clear about what you are doing.
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Last edited by v8archie; October 16th, 2014 at 05:18 PM.
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October 16th, 2014, 05:57 PM   #4
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Quote:
Originally Posted by Karma Peny View Post
Of all the areas of mathematics that I have encountered (school-level set theory, calculus, geometry, probability and so on) I could find no valid reason for using infinity.
It's true, if you don't study very far there doesn't seem to be much need for it. But it's an essential simplification for a great deal of work, like the real numbers.

Quote:
Originally Posted by Karma Peny View Post
For example, if we accept the infinite set of all natural numbers actually exists {1, 2, 3, 4…}, what is the percentage of odd numbers in this set?
The natural density of the set of odd numbers in the set of natural numbers is 1/2.

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Originally Posted by Karma Peny View Post
The obvious answer of 50% is problematic because it implies an even number of elements whereas infinity is supposedly neither odd nor even.
No, that's a misunderstanding of real numbers. Perhaps you can work this one out on your own since you've studied calculus.

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Originally Posted by Karma Peny View Post
Another option is to say the cardinality of the set of all odd natural numbers is exactly the same as the cardinality of all natural numbers, which is far from a satisfactory answer.
Why? For some situations cardinality is useful, for others natural density, for others there are other measures (logarithmic density, uniform density, etc.). It's a rich and beautiful system you're throwing out here!
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October 16th, 2014, 08:06 PM   #5
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Do we need the concept of Infinity in Mathematics and Physics?
One point of view :
We don't!
In the well-known standard system ZFC ('Zermelo,Fraenkel,Choice') of set theory the 'axiom of infinity' is well known to be independent of the rest. So we don't introduce inconsistencies by leaving it out. What we obtain is known as the system of 'hereditarily finite sets' . It is also known as the most simple non-trivial example of a 'Grothendieck universe'.

All programming languages which allow to create 'user defined types' (e.g. C++) allow to formalize the basic notions of physics and mathematics and do so obviously within the system of 'hereditarily finite sets (since, speaking C++, int and double are finite sets). Replacing type double by some 'multiple precision type' (like mp::real created by Pavel Holoborodko) one stays finite and neverthess can resolve all problems with 'numerical noise' in an experimental manner by increasing the number of bits which are used to represent reals.

So any argument that can be based on computer generated diagrams or animations is formulated per se without making use of infinity and limits (i.e. infinite processes).
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Last edited by Prakhar; October 16th, 2014 at 08:10 PM.
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October 16th, 2014, 08:07 PM   #6
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We find the limit as x increases. It does not 'tend to infinity' as this makes no sense. See the article on the Extreme Finitism website (the first link I mentioned) for my suggested notation and a full explanation.
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October 16th, 2014, 08:23 PM   #7
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Concepts like 'natural density', 'countably infinite' and 'transfinite numbers' are concepts that have been created to provide answers to problems that would not be there in the first place without the notion of a completed infinite set.

As described on the Extreme Finitism site, the weight of evidence is that completed sets of endless sequences (such as the natural numbers) cannot exist. Without the existence of infinite objects, these other concepts have no meaning.
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October 16th, 2014, 08:55 PM   #8
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Infinity, ad abstractum, exists.
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October 16th, 2014, 08:58 PM   #9
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Quote:
Originally Posted by Karma Peny View Post
We find the limit as x increases. It does not 'tend to infinity' as this makes no sense. See the article on the Extreme Finitism website (the first link I mentioned) for my suggested notation and a full explanation.
As x gets larger and larger the limit I suggested gets closer and closer to 1. But it never quite gets there. So how do you know it's really 1?

-Dan
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October 16th, 2014, 09:10 PM   #10
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Quote:
Originally Posted by CRGreathouse View Post
It's true, if you don't study very far there doesn't seem to be much need for it. But it's an essential simplification for a great deal of work, like the real numbers.
Real numbers include irrational numbers such as pi and the square root of 2.

The decimal expansion of an irrational number lacks precision, as the number can never be represented in its entirety. This is another example of where the notion of endless is often mistakenly taken to mean that a completed infinity can exist. As explained on my website, the concepts of endless and infinite are incompatible.
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