There's a tendency to get into all sorts of philosophical quicksand when talking about existence – does it mean physical existence, or space-time existence, is there a sense in which Harry Potter or Frodo have a fictional existence, etc. But let's keep it simple. When I ask whether the square root of 2 exists, all I am asking is whether there is a positive number such that when you multiply it by itself, the product is the number 2. In that way I haven't used the word 'existence' or 'exist' at all. Of course, I used the expression 'there is', but people don't seem to find the verb 'is' problematic in the same way as the verb 'exists'. So, is there a square root of 2?
This brought us to a discussion of sequences of decimals. We agreed that such sequences exist, i.e. that some things are sequences of decimals, or some sequences are of decimals, and we agreed that these sequences can be multiplied. Then we agreed that these sequences can be infinite, and that infinite sequences can also be multiplied. Finally, we agreed that (if we bought the other stuff), there is at least one infinite sequence such that when multiplied by itself, the product is 2 (or rather, the product is 1.999999… which you can verify on an Excel spreadsheet). So, there is a square root of 2. Or, if you like, the square root of 2 'exists'.
Is that it? Of course, we had to buy a couple of ideas. First, that some things are numbers. Some things are chairs, some things are tables, some things are stars or planets, some things are or may be angels (pace Anthony, who does not believe that any things are angels). And some things are numbers. Note my avoidance of the word 'exists'. Second, some numbers correspond to finite sequences of decimals, others to infinite sequences. Do we buy that? Time to read some more Ockham. More later.
11 comments:
Are we asking if there is a square root of 2, or are we asking if the square root of 2 is a number?
What things are or may be angels?
>> First, that some things are numbers.
Depends on the context of "some things".
As in some entities are numbers? I don't agree with that. Numbers are not entities. They are abstractions of attributes.
Are abstractions of attributes "things"? Sure. But only in the sense that "redness" is a thing.
>> Second, some numbers correspond to finite sequences of decimals, others to infinite sequences. Do we buy that?
Well, all numbers correspond to both finite sequences of decimals *and* infinite sequences of decimals. For instance, the number two corresponds to 2.0, to 2.0000..., and to 1.999...
Do any numbers correspond to only infinite sequences of decimals, and not to any finite sequences of decimals? Maybe. I haven't yet seen an argument which convinces me of this.
To start we need a definition of number, as a type of abstraction of attributes of entities, using a unit of measurement and a relationship between the unit and the thing being measured. Given that definition, to show that some numbers could not be expressed as finite sequences, we would need an example of an attribute of an entity (length, mass, etc.) which did not correspond to a finite sequence.
And by the way, I mentioned this in the previous post, but even once we have demonstrated that there are square roots of rational numbers which are not themselves rational (and I think it might be possible to get there), we still don't have the existence of the real number line, as there are only "countably many" square roots of rational numbers which are not themselves rational.
>> What things are or may be angels?
Well an angel is a messenger of God. I don't know why they have to be immaterial – I'll look it up.
>> Well, all numbers correspond to both finite sequences of decimals *and* infinite sequences of decimals.
That's interesting. Does the statement "1.9999 …. = 2.0" express an identity or an equality? If an identity, then an finite sequence of decimals is also a non-finite sequence, which is a contradiction. If an equality, then we would have to say what an equality was. (Frowns).
And after consideration of that definition, I am going to posit that yes, the square root of 2 is a number. Any measurement necessarily has an error bound. When we say that a stick is 2 feet long, we are saying that it is between 1.9999 and 2.0001 feet long, or whatever error bound is appropriate for the context. When we say that a stick is root 2 feet long, we are saying that it is between 1.41421 and 1.41422 feet long, or whatever error bound is appropriate for the context.
So the square root of two is an abstraction of the attributes of entities, using a unit of measurement and a relationship between the unit and the thing being measured. It is a number.
>> Does the statement "1.9999 …. = 2.0" express an identity or an equality?
I'd say it expresses a correspondence, a translation from one language to another, like "amarillo = yellow". (Is that an identity or an equality? I don't know.)
>> If an identity, then an finite sequence of decimals is also a non-finite sequence, which is a contradiction.
A finite sequence of characters ("1.999...") represents the same number as another finite sequence of characters ("2.0").
Put another way:
1/9 + 8/9 = 9/9 = 1
corresponds to
.111... + .888... = .999... = 1.000...
Someone mentioned this in the previous post, but...
Are you happy with the existence of the number 2, but believe that the existence of sqrt(2) needs to be justified?
The fundamental problems of "existence", whatever you mean by that, start with the integers. They have a mathematical definition too, if you want one. All the other numbers are built on top of the integers.
There is nothing terribly fundamentally interesting about "is there a number which, when squared, is 2" that isn't already in "is there a number which, when multiplied by 5, is 7".
At least, not from a mathematicians point of view. Infinity is nothing special, its just another property governed by axioms.
If you're working with the rationals only, then the answer to "is there a sqrt(2)"? is No. If you're working with the integers, the answer to "is there a number which, when multiplied by 5, is 7"? is also No (as long as you're not doing modulo...).
> Second, some numbers correspond to finite sequences of decimals, others to infinite sequences.
Not really. First of all (as I said elsewhere) we don't define numbers as decimal sequences. But they do correspond. But more, when you're talking about real numbers, 2_real is not the same thing as 2_integer (as I said above, 2_integer is actually a specific set (in the usual interpretation): {0, {0}} if I recall correctly. Whereas 2_real is more complex, as we know). 2_real -> 2.000...; its still an infinite decimal sequence (if you want to think like that).
>> Well an angel is a messenger of God.
And what is "God"?
>> Are you happy with the existence of the number 2, but believe that the existence of sqrt(2) needs to be justified?
Well, the existence of 2 has to be justified. But that's easy: 2 hands, 2 feet, 2 ears. Abstract from that to the number 2.
(You could also go with 2 meters, 2 inches, 2 yards, but then you're really dealing with a range around an error bound, and not something discrete.)
>> There is nothing terribly fundamentally interesting about "is there a number which, when squared, is 2" that isn't already in "is there a number which, when multiplied by 5, is 7".
True. We could have (and maybe should have) started there.
>> That's interesting. Does the statement "1.9999 …. = 2.0" express an identity or an equality? If an identity, then an finite sequence of decimals is also a non-finite sequence, which is a contradiction. If an equality, then we would have to say what an equality was. (Frowns). <<
Frowns indeed. With all 1.9... eyebrows. Both, I think. Certainly an equality, because the LHS can be seen as shorthand for an infinite summation expression which has the same value as the expression on the RHS. So the whole sentence is analogous, though infinitary, to 7+5=12. But also perhaps an identity because both 1.9... and 2.0 can be seen as distinct names that refer to a common value. Isn't the distinction between terminating and non-terminating decimals a syntactic distinction because it's a property of a graphical representation of a value which changes with the base of the representation. Thus in base 10 the value 1/9 has the non-terminating representation .1..., whereas in base 3 it has the terminating representation 0.01. On the other hand these representations (names?) can be calculated, so they contain arithmetic truths.
Is there a sharp line between syntax and semantics here?
>>But also perhaps an identity because both 1.9... and 2.0 can be seen as distinct names that refer to a common value.
Ah but then we lost the usefulness of referring to 'finite' and 'infinite' sequences. Indeed, is there any room for 'sequences'? The number is the number, the name is a finite string that can never be infinite, so ...
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