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What Kinds of Infinities Do You Find Philosophically Acceptable and Why?

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1. Do you believe that the set of natural numbers, N = {0,1,2,3,4,...} exists?

  • Reasons to disagree:
    • Some natural numbers are far too large and the universe is far too small or too short-lived to allow such numbers to ever be written down. This is called ultrafinitism.
    • All natural numbers exist, because any natural number could be written down in principle. The set as a whole does not, however, because you cannot write down the sequence of all natural numbers even in principle. This is called strict finitism.
  • Reasons to agree:
    • The set of natural numbers exists as long as it understood as potentially infinite. In principle, any finite sub-sequence of 0,1,2,3,... can always be extended by adding to it the next natural number in the sequence. In other words, an infinite set exists provided that we could, in principle, generate all of its members one-by-one, even though we cannot ever actually finish this process. This is called classical finitism.
    • The set of natural numbers exists because it can be defined. This is called Platonism.

2. Do you believe that the set of all infinite sequences of natural numbers, N -> N, exists? I.e., the set containing:

1,2,3,4,5,6,....

12,45,92,103,...

5,5,5,5,5,...

and so on...

?

  • Reasons to disagree:
    • I am an ultrafinitist. Since none of the elements of this set could ever be written down, none of them exist. And since none of the members of this set exist, neither does the set as a whole.
    • I am a strict finitist. Diddo.
    • I am a classical finitist. I believe that some members of this set exist because there exist finite programs which generate those sequences. Most of the members of this set do not exist, and so, neither does the set as a whole. However, because the set of all programs is potentially infinite, there is an alternative infinite set of sequences of natural numbers that does exist. This is the set of sequences of natural numbers that are computable. This is called effectivism.
  • Reasons to agree:
    • I am a classical finitist, but I do not agree that the human mind is limited to only computable operations. Given any finite sequence of natural numbers such as "1,56,987,23" the human mind is absolutely free to choose any natural number to continue such a sequence. In principle, this could be any member of the set N -> N whatsoever. As long as we understand the set of all such sequences to be potentially infinite, we could assent to its existence. This is called intuitionism.
    • I am a Platonist. Such a set exists because it is definable.

 

In addition to your own thoughts, where do you think Ayn Rand would fall? Rand believed in the existence of  potential infinities, so I think she would be a kind of classical finitist. But effectivist, or intuitionist? On the one hand, effectivist because I doubt she would assent to the existence of anything that could not have a finite representation in the form of a word. But on the other, she did believe in absolute free will, so she might have found intuitionism to be acceptable.

 

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1 hour ago, StrictlyLogical said:

Before getting into a set with an infinite number of members, just dealing with one or a few numbers, in the context of your question, what do you hold it means for “a number” or any “set of numbers” to exist versus not to exist.

Regardless of my personal opinion, I want to know what your answers to the above questions are as you understand the word "exists".

When it comes to abstract objects in general, I would consider myself an intrinsicist. I believe that certain abstract objects exist as part of the nature of reality. However, when it comes to mathematical objects, I believe that they are inherent parts of the nature of a certain part of reality, the mind. And further, that they exist if and only if they can be, in some sense, "conceived of", by the mind. Figuring out what counts as a "legitimate" conception is tricky. I am not completely decided about effectivism vs intuitionism. I have a far superior understanding of classical (Platonist) mathemtics as opposed to effective mathematics, and a superior understanding of effective mathematics as opposed to intuitionistic mathematics. However, I find myself leaning more and more towards the intuitionistic side every day.

But, coming back to philosophy now, I also do not believe that mathematical concepts are abstracted from concretes. Rather, I believe that they are necessary components of anything that can be called consciousness. In short, no math = no consciousness (at least not human consciousness).

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Part of a related article:

Elizabeth Pyatt's Thoughts on Cognition, Linguistics, Learning...Whatever

The Language Without Numbers

By ELIZABETH J PYATT on July 23, 2008 4:54 PM | Permalink

An interesting news story from the past few years is the Amazonian language Pirahã which lacks number words. That is, instead of counting quantities (1,2,3,4...), the Pirahã only estimate quantities (relatively small, relatively large). The latest study from MIT seems to confirm this. Interestingly, when objects are taken away from a pile, the estimates change so that "small" may become 5-6 instead of 1-2 as previously thought.

This has perplexed linguists since almost all languages have some sort of counting (even in remote locations). The only other examples of low-tech numbers had been systems of 1,2, many. We normally think of counting as a "basic" cognitive skill, but it appears to be primarily cultural.

I first about this in 2000 from a guest speaker Peter Gordon. His evidence was convincing, but there have been some points I have been pondering.

  • Pirahã children who learn Portuguese also learn to count - it's not a difference in cognition [Peter Gordon, personal communication]
  • Not surprisingly, male laborers in Brazil are stiffed a lot because they do not pay as much attention to "exact" quantities. However the Pirahã women are reported to gently mock their men folk for this [Gordon, p.c.] It reminds me of cultural gender stereotypes like men can't pick coordinating colors and women can't work with computers (and yes many of us buy into them whether they are 100% true).
  • Many animals can easily distinguish quantities of 1,2,3, (or a little more) on sight, but after that they guesstimate. In this study, monkeys can recognize quanities of 1-4, but estimate after that. This predicts that a basic counting would be something like 1,2,3, many, but the Pirahã system is even more basic.
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One important point to get clear is that individual natural numbers and all the other mathematical things mentioned are mental constructs.  They definitely exist as mental constructs.  They do not exist as physical objects.  They do not exist as entities in some Platonic heaven nor as Platonic forms nor as Aristotelian essences; there are no such things.

Mental constructs exist as such even if they are not valid.  Examples are the concepts "extremist" and "God".

 

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6 hours ago, Doug Morris said:

One important point to get clear is that individual natural numbers and all the other mathematical things mentioned are mental constructs.  They definitely exist as mental constructs.  They do not exist as physical objects.  They do not exist as entities in some Platonic heaven nor as Platonic forms nor as Aristotelian essences; there are no such things.

Mental constructs exist as such even if they are not valid.  Examples are the concepts "extremist" and "God".

 

What kinds of infinite sets do you think are valid?

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