Thursday, April 12, 2012

What is the continuum?

There's a discussion going on here about how Aristotle defined 'the continuum'. The problem, of course, is that he didn't, and couldn't, define the English word 'continuum', since he wrote in ancient Greek. A further problem is the English word is imported from medieval Latin. Is the English imported sense the same as the medieval sense? Even assuming that the medieval Latin was an accurate translation of Aristotle's Greek, how far does the modern usage of the word reflect the medieval usage?

The modern use, as I understand it, is as an abstract noun referring to an abstract non-physical entity with certain idealised properties. This contrasts with the medieval use we find, e.g., in Aquinas here which retains the sense of the adjective, namely as signifying that (physical thing) which has continuity, rather than the abstract feature of contuinity itself (whatever that means). 'Continuum' in Latin, like 'vacuum' is an adjective in the neuter which (in that usage) has a noun-like sense, meaning 'the continuous', or 'that which is continuous' or 'that which is unbroken'. E.g. when he says that it is impossible that "aliquod continuum componi ex indivisibilibus" he is not saying that it is impossible for some abstract object called 'the continuum' to be composed of indivisibles. Rather, he is saying that it is impossible for any real object possessing the property of continuity or unbrokenness to be composed of indivisibles.
Dicit ergo primo quod si definitiones prius positae continui, et eius quod tangitur, et eius quod est consequenter, sunt convenientes (scilicet quod continua sint, quorum ultima sunt unum: contacta, quorum ultima sunt simul: consequenter autem sint, quorum nihil est medium sui generis), ex his sequitur quod impossibile sit aliquod continuum componi ex indivisibilibus, ut lineam ex punctis; si tamen linea dicatur aliquid continuum, et punctum aliquid indivisibile.
Now immediately, hearing this, there will be those who cry that Aristotle was thinking too hard about the 'real world' or the 'physical world' or something like that. As opposed to the 'mathematical world' or some abstract world of abstract things. To which I confess: I don't understand. If there is a mathematical world, in what sense is it not real? As for abstraction, I commented earlier (somewhere) that abstraction is considering normal, real things without considering the features which we are abstracting from. For example, while there is no such thing as a frictionless surface, I can still consider surfaces in respect of their shape and form, without considering properties such as friction. That is all that abstraction is. Or I can consider a triangle without considering whether it has (A) all three sides equal, or (B) two sides only, or (C) none. Now any triangle I consider must be one of (A), (B) or (C). Yet I can consider any one of them without considering whether it is such, i.e. in abstraction from whether it is any one of those three types. That is all 'abstraction' means. It doesn't mean there are any such things as 'abstract objects', as though, absurdly and impossibly, there could be a frictionless surface, or a triangle which does not have three sides equal, nor two side, or none.

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24 Comments:

Blogger Anthony said...

Excellent post. I wonder whether you were just trolling (playing devil's advocate) with your other posts on this topic.

Yes, Aristotle didn't define the word "continuum", because he didn't use the word "continuum". The argument you presented after "Aristotle's argument is this" is not in fact Aristotle's argument at all.

11:40 am  
Blogger Anthony said...

As for the term "abstract objects", I guess I haven't yet developed the vocabulary to discuss my thoughts on the term. The difficulty lies in what are referred to as "abstract particulars".

I say "referred to as" because I see the term as contradictory - to engage in abstraction is to consider the common feature among more than one. And the common feature among more than one is a universal, not a particular.

However, there are certain things which are referred to as "abstract particulars", and these are where the difficulty lies, or at least where I have not yet worked through the specifics.

One example would be the number 2. It seems to me that the number 2 is a universal, not a particular. 2 ears, 2 eyes, 2 legs, 2 feet... We abstract from these instances of twoness and arrive with the number 2. We then treat 2 as though it is a noun and not an adjective, and this is quite convenient. We say "2+2=4", but what we really mean is that two of something plus two of something equals four of something. (I really see no reason this is any different from any other universal. "Red chair, red desk, red cup", then the adjective is used as a noun, and we have the universal redness, and can say "red + blue = purple".

More difficult is something for which there is only one instance, or for which there is no instance. For example, a sentence is considered an abstract particular. There may be only one or zero instances of that sentence. So if a sentence is a universal, from what multiple instances is the universal being abstracted?

At this point my vocabulary starts to fail me. I think the answer is that the instances from which the universal is being abstracted are "intensional objects". But I could be abusing that term. In any case, a sentence is abstracted not necessarily from currently-existing objects, but from potential objects. I write a sentence once. It is embodied in physical form only once. But I can potentially copy the sentence over and over, and I abstract from the qualities of those potential copies.

1:18 pm  
Blogger Edward Ockham said...

>>One example would be the number 2.

Beyond the scope of this post, I'm afraid.

1:57 pm  
Blogger Belette said...

http://web.maths.unsw.edu.au/~jim/AristotleContinuum.pdf says "Since a continuum is “that which is divisible into parts which are always further divisible” (Phys. 232b25),". That would apparently make it a matter of definition that the continuum can't be composed of indivisibles. If so, I don't understand why he needs a proof. That link was found by NS; see http://scienceblogs.com/stoat/2012/04/aristotle_and_the_continuum.php.

> he is saying that it is impossible for any real object possessing the property of continuity or unbrokenness to be composed of indivisibles

How does this help? You've replaced the undefined word "continuum" with the undefined word "continuity".

> If there is a mathematical world, in what sense is it not real?

Well, you can think like that if you like, though it seems odd (indeed, incomprehensible) to me. In which case I suppose you're accepting the mathematical "real line" as part of the real world. In which case you're accepting an object definitely composed of points as continuous (err, assuming the maths meaning is close enough to A's undefined meaning), and therefore as a counter-example to A's proof.

2:41 pm  
Blogger Belette said...

s/NS/NB/

2:41 pm  
Blogger Jason Hills said...

Clarification on something you were thinking I I don't believe you've said, E. Ockham. A number of the comments seem motivated by more contemporary notions of continuity or the continuum, or at the very least mine were. Whether that is true of Aristotle is another matter, and it is good that you refocus and stick to your guns.

3:20 pm  
Blogger Belette said...

> You've replaced the undefined word "continuum" with the undefined word "continuity".

Quick test of the meaningfulness of *your* (or A's) definition of continuous, or continuum: are the rationals continuous (under A's / your definition)? If not, why not?

3:28 pm  
Blogger Edward Ockham said...

Belette: I am generally not seeing where you are coming from.

3:51 pm  
Blogger Jason Hills said...

"Continuity" is a well-defined term in recent mathematics and logic. Many of your comments, E Ockham, would not apply to those concepts. I have been hinting at that, and perhaps you didn't pick up on that. B is making this explicit, I bet.

3:53 pm  
Blogger Jason Hills said...

http://mathworld.wolfram.com/Continuity.html

3:54 pm  
Blogger Edward Ockham said...

ah this http://scienceblogs.com/stoat/2012/04/aristotle_and_the_continuum.php makes it clearer. Time for another post.

I concede there is an error in my earlier presentation of A's argument. He is not asking whether the continuum is composed of indivisibles, but rather whether a *line* is composed of etc. That still leaves the question of what he means by 'line'.

Jason: you are making good points and I am not ignoring them, but still thinking about them. Thank you for the hints.

3:58 pm  
Blogger Jason Hills said...

Thank you for your kind words. What I bring up is not necessarily relevant to your actual discussion, i.e., about Aristotle, and I had presumed that the distinction between ancient and recent notions was obvious, but then I have a degree in mathematics and have perhaps a skewed perspective. As I was reading many of your posts, I kept wondering about infinitesimals and perhaps Newton's discussion of them (that I know about but not much more).

4:14 pm  
Blogger Belette said...

Hopefully you now know where I'm coming from. If changing from "continuum" to "line" makes any difference then yes I suppose you need a new post with (as you say) the need to define "line".

In which case the "quick test" of the definition would be "do the rationals form a line under your definition? If not, why not?" Ditto for the reals.

5:54 pm  
Blogger Belette said...

In case it helps, I think that "points only come into (actual) existence for Aristotle when a division is made between two line segments" is the explanation for his problem (from the pdf I quoted earlier).

6:36 pm  
Blogger Anthony said...

>> If there is a mathematical world, in what sense is it not real?

>> Well, you can think like that if you like, though it seems odd (indeed, incomprehensible) to me. In which case I suppose you're accepting the mathematical "real line" as part of the real world.

Or he's not accepting that there is a "mathematical world".

Or he's not accepting that the mathematical "real line" is part of the "mathematical world".

(I'm not sure what exactly he means. The sense in which there is a mathematical world which is not real is the same as the sense in which there is a Harry Potter World (or any other fictional world) which is not real. That is to say, that "there is a mathematical world" is not meant to be taken literally. However, I don't believe "there is a mathematical world" is something I have said. Certainly I believe that much of modern mathematics is fictional.)

12:13 pm  
Blogger Anthony said...

>> In which case the "quick test" of the definition would be "do the rationals form a line under your definition?

Certainly not under the definition of Aristotle. A line connects points, not numbers. (I know Ed seems to have implied that Aristotle said otherwise, but that seems unlikely, and I can't find it.)

If you assigned colinear points rational numbers, based on their distance from an origin point, you could then connect those points with a straight line.

12:26 pm  
Blogger Anthony said...

Aristotle: "that which is intermediate between points is always a line"

So, there's his definition of line. In modern terminology, this would be called a "line segment".

12:41 pm  
Blogger Edward Ockham said...

I'm simply saying (a theme of many of my earlier posts) that I only accept what is real. Whatever exists = what is real. What doesn't exist we don't need to worry about. We don't need to worry about Harry Potter e.g. (Of course, we might worry about the way the author has set up the plot, or about the author's spelling and grammar, or the quality of her writing, for all that is real, and possibly relevant).

2:51 pm  
Blogger Anthony said...

"What doesn't exist we don't need to worry about. We don't need to worry about Harry Potter e.g. (Of course, we might worry about the way the author has set up the plot, or about the author's spelling and grammar, or the quality of her writing, for all that is real, and possibly relevant)."

I find it interesting that you would say that. I very much agree with it, but I seem to remember us having a bit of a disagreement over the importance of determining the meaning of fictional passages. (It was during the whole Superman discussion.)

I can't find the comment, but I remember you saying something that I took as extremely counter to the position you are now taking.

Ah, here it is:

Me: "If all fictional statements are false, what's the point of studying their meaning? "

You: "That's a bizarre question, but let me answer it anyway. Explaining how they have a meaning (at least for singular fictional propositions) is a challenging question, and therefore the question is worth studying. According to one major theory of meaning, singular fictional propositions (i.e. declarative sentences) don't have a meaning at all."

11:52 am  
Blogger Anthony said...

http://ocham.blogspot.com/2012/01/reprise-reference.html

12:00 pm  
Blogger Edward Ockham said...

I think the positions are quite consistent. I am saying that (singular) fictional statements are meaningful, counter to the DR view that they are not. But I am saying that, because they are false, and because they are not actually talking about anything - they are not worth worrying about.

By contrast, I think some mathematicians here think that in some important sense the mathematical statements, while genuinely fictional, are in some sense 'true'. Summarising:

Direct Reference: Fiction is meaningless
London Ockhamists: Fiction is false
Certain mathematicians: Fiction is true

12:14 pm  
Blogger Anthony said...

Anyway, getting back on topic, does anyone following this thread have an explanation of what in reality corresponds to the real number line?

To put it another way, is there anything in applied mathematics which deals with the uncountably infinite?

12:24 pm  
Blogger Anthony said...

Ed, hopefully I remember to come back to your comment about fiction when I have more time to think about it.

Hopefully by then you can answer this: what is the difference between "meaningful" and "worth worrying about".

12:27 pm  
Blogger Edward Ockham said...

Well something can be meaningful without us having to worry about what it means. For example, the expression "not worth worrying about" is meaningful, you agree. But what it means (or signifies) is something not worth worrying about.

12:42 pm  

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