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[Edit 08 March 2016: Some of my views have changed or evolved since I posted this thread and responded to the posters in it. These have been added to the essay in a new section.]

This is a re-post from the mathematics section, where the question appears to have been off-topic. The title of this question is self-explanatory, and I have posted my opinions on it in a little 7,000-word essay on my website: http://www.thomaswhichello.com/?page_id=372. I'd appreciate hearing your views on them. The chain of reasoning and proofs I have devised are too long to be condensed here, although to present the crux of my belief-system in brief it is this:

First, that "Decimals are simply specialized adjectives of comparative size which define an implied natural number. They are not themselves a new or intermediate class of number, and, indeed, no such class of intermediate number between the natural ones exists."

Secondly, that, as decimals actually imply nothing more than natural numbers, therefore there cannot be any discrepancy of infinity between the set of all natural numbers and the set of all decimals.

An excerpt:

"Let us say that I have a piece of cake, that is to say, one piece of cake, and I divide that piece of cake into four equal portions. I then take one of those portions. I now have in my hand one piece of cake—just as surely as the original is one piece of cake. I may, at this point, choose to analyze the physical size of that piece of cake in my hand by comparison with the piece of cake from which it was taken. This is 1/4th of it expressed as a fraction, or 0.25 of it expressed as a decimal. And yet this new piece of cake that I hold in my hand remains, numerically, as I say, one piece of cake. Yet what, therefore, does the fraction or decimal represent? What it represents is, that this particular piece of cake that I hold in my hand, although numerically one, is worth, in point of its physical size, one piece of cake out of the original piece of cake from which it was taken, once that original piece of cake has been divided into four equal portions. That new piece of cake which I hold in my hand, the decimal or fraction explains, is so much smaller in point of size as compared with that other, original piece of cake. But numerically, they are both one—both the piece I have taken, and the original which I divided up before I took it from it.

A further physical example that may more accurately demonstrate how the decimal or the fraction functions in the abstract realm, is if the reader first imagines me to have a piece of cake all on its own; secondly imagines another, larger piece of cake; and lastly is given to understand that the piece of cake I have in my hand, by virtue of sheer coincidence, is worth, in point of size, exactly one quarter (or what have you) of the larger piece of cake, though I had not taken it from that cake—simply by comparing the one thing with the other thing, and not by taking it out of it. And in this case, again, we might describe that piece of cake in my hand as being worth 1/4th, or 0.25, of the other, larger piece. This analogy, besides being more accurate than the former in its relation to pure mathematics, highlights the absurdity of calling a fraction or a decimal a number even more greatly. That I choose to compare the size of an object in my hand with that of another object does not affect the numerosity of my object, the fact that I continue to hold one object; and that is all that a decimal or a fraction does."

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    I want to point out one thing about your paper. The rational numbers (which includes all decimals that can be written in fractional form... ie: repeating decimals) is countable. ie: you can form a one-to-one correspondence between the natural numbers and the rational numbers. The real numbers (which includes irrational numbers like squareroot(2) is not countable. In your paper you don't mention irrational numbers at all. Plus, I don't think anyone doubts that the natural numbers are somehow primary, and decimals are higher level abstract objects. We call them all numbers for convenience. – Ameet Sharma Mar 5 '16 at 16:15
  • Your idea is not without precedent. Others have pursued the idea that the irrationals are not actual numbers but just limits. The irrational value of pi, for example, might be described as the limit of the Taylor series expansion of arccos(-1). Certainly, any decimal expansion that you write down, or any quantity that you can measure, or store in a computer, is only a rational, because it has a limited degree of precision. But most mathematicians, like Hilbert, do not want to be expelled from Cantor's paradise, so this idea has not achieved widespread acceptance. – Bumble Mar 5 '16 at 16:31
  • @TW294287, I think there are 2 separate matters here (practical vs metaphysical). I think the value of Cantor's arguments are within mathematics (as a practicality). The question of what type of infinities "actually" exist in the real world is a different matter. To connect arithmetic and geometry/calculus, you need the real numbers, at least within the abstract realm of mathematics. And within that abstract realm, there's no doubt that we can't form a one-one correspondence between the real numbers and natural numbers. – Ameet Sharma Mar 5 '16 at 16:42
  • @TW294287 What is your background in mathematics? – Jo Wehler Mar 5 '16 at 17:01
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    You said something brilliantly true in your essay. "Indeed, I know virtually nothing about mathematics." – user4894 Mar 5 '16 at 18:41
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Set Theory proves that there are multiple infinities of sets. I don't see how showing that decimals (reals?) are not numbers has anything to do with it. Sets are the only objects in Set Theory; what you "do with them" later on, such as using them to define the natural and real numbers, simply does not matter for the purposes of the argument.

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  • @TW294287 But still, the power set of the set of natural numbers (for example) would be of a greater cardinality than that of the set of natural numbers. And there are even greater cardinalities (the power set of the power set, etc). – Eliran Mar 5 '16 at 18:59
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    @TW294287 : I see. So your argument starts by denying that sets can be infinite, and ends by concluding that any two infinite sets have the same cardinality. That shouldn't take much argument. Once you deny the existence of unicorns, it's pretty easy to prove that all unicorns are the same color. So why all the verbiage? – WillO Mar 8 '16 at 18:22
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    Cantor's entire proof has nothing to do with numbers, so whether or not some specific thing is or is not a number cannot change whether or not this argument which is not about numbers is true or false. Cardinality is a mapping both ways that does not map elements onto the same target. Whether you consider that 'counting' or not does not change the meaning of Cantor's proof. – user9166 Mar 9 '16 at 0:28
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Comments

1)

The so-called natural numbers, that is, one, two, three, etc., are said to be countable, which means that they can be put in a one-to-one “correspondence” with other “sets” that contain elements which are also associated with the natural numbers.

It is the other way: a set is countable if it can be put in a one-to-one “correspondence” with the natural numbers.

We assume as "primitive" the fact of counting, i.e. ordering one by one the elements of a set.

2)

the so-called real numbers, [...] that is to say, [...] the so-called decimal numbers.

Real numbers are not defined through their decimal representation; ancient Greeks discovered the existence of irrational magnitudes (the diagonal of the unit square) proving that we cannot find two natural numbers r and s such that the diagonal of the square of side 1 can be expressed as the ratio (fraction) r/s, i.e. what we today call rational numbers.

Thus, having proved that there are magnitudes that cannot be measured by rational numbers, mathematician assumed the existence of irrational numbers.

The set of real numbers is formed by the set of rationals (which include the fractions n/1, i.e. the natural) and the set of irrationals.

3)

For is not infinity infinity?

Cantor showed that, with a suitable new way of "comparing" the size of sets (equipotence) we have mathematically well defined different "levels" of infinity.

The fact was already "intuited" by Medieval philosophers and discussed by Galileo: see Galileo's paradox.

Cantor gave precise mathematical definition to this intuition.

4)

Cantor claimed that, by forming a new set of all the “subsets” of an original “infinite” set, we may make a “greater” infinity than that of the original set.

Cantor proved that, according to his newly defined method of comparing the size of sets, the set of all the “subsets” of an “infinite” cannot have "the same size" of the original set. Thus, due to the fact that the first set is a subset of the second one, the latter must have "greater size" than the former.

5)

The fallacies inherent in Cantor’s findings about infinity must be refuted in two ways. First, by understanding that so-called real numbers, such as decimals and fractions, are not numbers.

The so-called real numebrs are not defined by their decimal representation and they are not fraction.

But - as said above - the existence of real numbers is in a sense assumed: we (i.e. quite all the mathematicians and scientists since at least the Early Modern Era) want that there are numbers to measure all the magnitudes, also those proved "irrational".

6)

Any of the real numbers which is in-between the integers can, in turn, be represented, in all cases by a decimal representation, in some cases by a fractional representation.

No; no fraction con represent the square root of two.

7)

what, therefore, does the fraction or decimal represent? What it represents is, that this particular piece of cake that I hold in my hand, although numerically one, is worth, in point of its physical size, one piece of cake out of the original piece of cake from which it was taken, once that original piece of cake has been divided into four equal portions. That new piece of cake which I hold in my hand, the decimal or fraction explains, is so much smaller in point of size as compared with that other, original piece of cake. But numerically, they are both one.

Of course one piece is one peice. If you count the four peices of cake you "count up to four" and this of course does not mean that you have four cakes.

But with fractions you do not count objects; you measure magnitudes: lenghts, areas, wights.

I'm sure that you can have some difficulty in convincing a boy that a full cake and one-tenth of it are the same, because both are one...

8)

Pure mathematics is so much concerned with the abstract that it can be easy to lose sight of the realities on which it ought to be based, [...]. In the end, mathematics ought to be a representation, first of reality, next according as that reality is extrapolated into the abstract realm in accordance with reason.

Exactly; from ancient time humans found "continuous magnitudes", like lines and they searched precise way to measure those magnitudes; thus the discovery of the irrationals: what number may "measure" the lenght of the diagonal of the unit square ?

9)

As abstract numbers and nouns themselves [...] exist in a potential state, for they exist only under the presumption that they might potentially be filled in with some concrete example [...]. Thus in the abstract, if I add one to one, I get two; if I add two to two, I get four; but we may supply any number of concrete objects in the place of these abstracted numbers and they will work just as well.

Yes, the basic "intuition" grounding the natural numbers is the possibility of indefinitely iterate the operation of adding one. We may imagine to go on ad libitum with the "game" of +1, and thus we are unable to set an upper limit to the natural numbers.

This means that there are an infinity of objcets in the univers ? We do not know, and here lays the "mathematical abstraction".

With the real numebrs, the "grounding intuition" is our "image" of the continuum of the real number line: where to stpo the process of dividing a finite segment in two parts ?

10)

In modern mathematics, a rational number r/s is defined as the pair of naturals: (r,s).

How we "compare" them ? Ordering them and reducing the comparison problem to the well-know ordering of natural numbers: we say that (r,s) < (p,q) exactly when rq < sp.

This means that:

2.1 is bigger than 2

because we can "compute" the fact that 2 < 2.1: 2 x 10 = 20 < 21 x 1 = 21, i.e. (2,1) < (21,10).

Having defined rationals (the former "fractions") as pair of naturals, Cantor's method of "comparing" the size of sets can be applied to prive that the set of naturals and the set of rationals have the "same size".

11)

In a similar way, modern mathematics define reals as subsets of rationals.

But Cantor proved that the set of all subsets of a given set cannot have the same size of the original set: it has a "greater size".

Thus, having showed that the set of reals has the "same size" of the set of all subset of naturals, it follows that it has a "greater size" than the set of naturals.

The set of naturals has a countable "size"; thus, the set of reals has a "more than countable size".

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  • @TW294287, Please define what you mean by "number". – Ameet Sharma Mar 5 '16 at 17:47
  • @TW294287, ok but how does it follow that "descriptions of comparative size" and "natural numbers" are the same classes of object. They're two different things. If real numbers are "descriptions of comparative size", then there are more "descriptions of comparative size" than there are "natural numbers". – Ameet Sharma Mar 5 '16 at 17:54
  • @TW294287 - the fact that the sqaure root of 2 is definitely not identified by any finite decimal representation approxiamting it, has been proved 2,5 thousand years ago and I'm not interested to discuss it... – Mauro ALLEGRANZA Mar 5 '16 at 17:56
  • @TW294287 - I agree that the philosophical issues regarding infinity are slippery and I'm not asserting that mathematics has solved them. But the "countable-uncountable distinction" is mathematically meaningful and interesting and in no way we will throw away math simply because it is not able (but who is ?) to answer all the relevant philosophical problems. – Mauro ALLEGRANZA Mar 5 '16 at 18:16
  • @TW294287, Do you know about functions and one-to-one correspondence? Do you agree that the real numbers cannot be put in one-to-one correspondence with the natural numbers? – Ameet Sharma Mar 5 '16 at 18:26
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With all the linguistic arguments you put forth, you should be able to accept an argument that what you have actually done is define "number" in a way which defines something which is distinctly different from what others use the word "number" to describe. As a result, the acceptance of your argument will be heavily based on how useful it is for handling linguistic situations others may come across.

In my opinion, there are some interesting challenges it faces for example:

But what is a decimal? And what is a fraction? Analyze them and you will find that they are, in fact, not numbers, but adjectives.

Implied in the word in is that number are not adjectives. Presumably they are nouns, which is how most people treat that concept. However, now we have a sticky question of defining operations on these numbers. Consider division. "Six divided by three equals two," or in expression form, 6 / 3 = 2. One can divide two numbers to get a number. However, now consider "Three divided by two equals one point five," or in expression form, 3 / 2 = 1.5 The former is an operator that takes two nouns and produces another noun. The latter, by your definitions, is an operator that takes two nouns and produces an adjective. Likewise "One half plus one half equals one," or 0.5 + 0.5 = 1 took two adjectives and combined them to be a noun! While you can define operators to do all sorts of interesting things, when you are defining the meaning of words, linguistically people prefer to simplify away complexity when it does not provide enough value. In this case, people have found it very effective to think of decimals and fractions as nouns. Just think of how hard it would be for algebra to work if x+y yielded either a noun or an adjective, but you don't get to know until you finish solving the equations! In fact, they think of it so much so that they do not find many inconsistencies with that assumption at all. (One rare counterexample I know of is the infinite decimal 0.9999... is equal to 1, which is an odd quirk that does come up with infinite decimals).

I include the irrational numbers when I say decimals. I know that they are called numbers out of convenience, but it is this casual error, I believe, which has led us all into making the mistake of conceiving that there are multiple degrees of infinity.

This is not from your essay, but from a comment you made above. It is very difficult to include irrational numbers in decimals without needing to be able to construct infinite series. The rational numbers "behave" much better than the irrationals in many scenarios. For example, the issue I mentioned above with 0.99999... equaling 1 is an issue that only raises in irrational numbers. This sort of strangeness never occurs in rational numbers.

In the end, your opponents for this essay is proof theory, the Peano axioms (for natural numbers), and their realization in set theory. These three concepts are very very powerful concepts in mathematics. The multiple infinities that both you are a straight forward side effect of those basic assumptions. To refute the idea of multiple infinities is to reftue one of those fudamental cornerstones of mathematics. In that sense, your articles is not "A Refutation of Georg Cantor’s Findings about Infinity," but rather "A Refutation of All of Mathematics (or at least most of it)." Needless to say, this is quite the bold challenge, and your thesis will have to prove great implications before its value exceeds that of all of mathematics.

Which leads us back to where we started. As Bumble pointed out, you are not the first one to question the canon definition of infinity. Some, such as the finitists, don't even believe infinity exists! However, when one looks at this from a linguistic perspective, this means the concepts encoded in your words will always be slightly different than the concepts similar words would encode when said by others. You are welcome to say "In my wording, a fraction is an adjective, not a noun." However, if you try to say "You are wrong, because fractions are adjectives, not nouns," you have attempted to force your linguistic constructs on someone.

Your discussion of the philosophy of mathematics may benefit from actually learning more mathematics. For example, you may see that the mathematical community does not always agree on their axioms. For example, there is an axiom known as "The Axiom of Choice." In laymans terms, the idea of the axiom of choice is that you can have a bunch of bags containing objects, and in all cases you can construct a new bag that contains one object from each of the old bags. This seems obvious. You just go to each of the old bags, pick an item out of each, put it in the new bag. However, when you start playing with infinities, it gets murkier. So murky, in fact, that the Banach–Tarski paradox shows that, if you accept the axiom of choice (AC for short), you can take a sphere, cut it up into a finite number of pieces, and reassemble them into two dense spheres (no holes), with each sphere having the same volume as the original! Duplication!

The result of this is that AC is not accepted by everyone. This duplication issue bothers many people enough that they will not accept AC as part of a proof. Set theorists who do not believe in AC who face a proof that demands AC will not say, "Oh, this theorem isn't valid. The mathematician proving it made a mistake." They will say "Oh, this mathematician assumed the axiom of choice. I do not accept this, but I can say that, if you accept the axiom of choice, the theorem is proven."

Likewise, you are free to develop you own concept of mathematics, and bring it up to a state where others find value in it, but you will not be able to get away with saying "Oh, that statement it false" when what you really mean is "That statement relies on assumptions I do not accept, but if you do accept those assumptions, the statement is valid."

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  • @TW294287 Might it be more logical to, instead of claiming fractions are numbers but adjectives, to instead explore that all numbers might be adjectives? – Cort Ammon Mar 5 '16 at 19:28
  • Do you have any axioms in mind that you might challenge? Many, such as the peano pxioms are really hard to refute philosophically – Cort Ammon Mar 5 '16 at 19:31
  • Have you looked at finitist reasoning? They have similar concerns. However, being mathematicians, they have sought to actually identify where adjustments should be made in the way we think about mathematics. For example, they have to consider how to carefully construct Calculus so that it still works within their framework, or explain why it doesn't work (which is tricky because it is so successful at describing the world around us). Constructivism in mathematics may be another interesting place to look. – Cort Ammon Mar 5 '16 at 19:47
  • Is it possible that the abstract mental realms of others might contain multiple kinds of infinities when yours does not? How confident are we ever that we understand someone's mind? Alternatively, I may recommend looking at Category Theory. Most mathematicians accept abstract mental concepts of infinity from set theory, with the axiom of infinity, which you seek to refute. If you are not finitist, then you have to be able to arrive at an infinity via another approach. From what I undersatnd, Category Theory can serve as an alternate foundation for mathematics, because you can define... – Cort Ammon Mar 5 '16 at 20:09
  • ... set theory from its constructs. You may find that you can fit an infinity into Category Theory in a way which is more aligned with the way you are thinking, and simply not define a set theory with the axiom of inifinity in it. Category theory has a very different feel from set theory, so it may align better for you. – Cort Ammon Mar 5 '16 at 20:10
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Well, one could simply negate the axiom of infinity in set theory; and people have done, this is one version of the finitist programme; another party even object to very large numbers, they belong to the ultra-finitist faction.

Still, the main-stream opinion is to keep this axiom.

It's worth remembering that decimal are representations of numbers, not the numbers themselves; so one third is the same as 1/3, which is the same as 0.33333...; and in fact I could arbitrarily draw up a new sign that represents this number - like ¥ - except it messes with the regularity and order that we expect with mathematical notation.

So, yes; you're right about decimals - where I'm using the word representation, you're using the word adjective - it amounts to the same thing.

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