Catholic and Protestant Spitituality

[AllanS]

[Kolbitar]

:: Apple isn't a meaningless term, it's an approximate term. "Apple" refers to a set of unique objects with sufficient similarites for them to be given the same name. At the edges of this set, you'll have people saying "That? You think _that's_ an apple? It's not the right shape and it tastes like crepe." And others will disagree.

Hey Allan.

I would ask, sufficent similarities? or sufficent identities? All things have certain aspects identically in common with other things (aside from spacial location). To begin, all things have in common the identity of existence, of thing-ness. From there they can be matched with other things having sufficent identities which define them as this or that kind of thing. I don't think that the limits to our knowledge, our inability to categories certain things, refutes basic facts like some apples are definitely apples.

[AllanS]

[postodave]

Well, now, we've stumbled onto a discussion of Philosophy of maths. Bearing in mind that I am not a mathematician I do find this facinating. Firstly I can take one thing and put another next to it and demonstrate 1+1 making 2 on a particular occasion but the statement 1+1=2 is supposed to be universal. I cannot demonstrate it always making two. Hence 1+1 =2 cannot be demonstrated. J. S. Mill argues that maths is inducted from sense experience. Certainly it cannot be proved. Once upon a time back in the nineteenth century a project began to reduce maths to logic. Frege tried to do this using class theory. He begins by defining 0 as the class of classes that has the same number of members as there are objects not identical to themselves. Then because there is only one number zero he can define 1 and now he has two numbers so he can define 2 and so on to infinity. He wrote a book expressing arithmetic in terms of logic using class theory and beggining from self evident axioms. But while it was still being type set Bertrand Russell wrote him a letter pointing out that some of his self evident axioms were contradictory. Then Russell working with Whitehead had a go at doing it himself. Anthony Kenny made a very witty comment about this book (Principia Mathematica) Russell had criticised Aquinas for trying to use philosophy to prove things he already believed like the existence of God. Ah well says Kenny Russell wrote this huge book trying to prove that 1+1=2. Then in 1931 Kurt Godel demonstrated that maths could not be reduced to logic. Any logical system that contained all of number would have undecided statements.

As for other kinds of maths I genuinely think that working in different bases you are only changing the way you symbolise numbers not the actual operations you are symbolising. And in other forms of math like boulian you are really doing different operations.

After proving maths could not be reduced to logic Godel (he should have an umlaut but frankly I can't be bothered to write it) became a kind of platonist. He believed numbers exist in a seperate realm. John Polkinghorne uses this as a basis for arguing the existence of a noetic realm of creation alongside the material wherein subsist not only number but perhaps also such entities as Paul's principalities and powers, Jungiam archytypes and even noetic life forms (angels)

We don't know what numbers are but we experience them in our daily lives. In that sense very like God! :D

[robsia]

[robsia]

[AllanS]

Hi Dave (is it Dave?),

"Firstly I can take one thing and put another next to it and demonstrate 1+1 making 2 on a particular occasion"

No you can't, because no two objects are identical. Am I not seeing something here? An apple plus an orange don't equal two apples unless we pretend they're both apples. We can only count by pretending things are identical, which is to not see them as they actually are.

It's the same with all language. "The cat sat on the mat" seems to have meaning, but only because I pretend cat, sat and mat mean the same to me as they do to you. But which cat? A lion, a Persian, a kitten? Sat how? Feet together? Tail under or behind? Which mat? and so on. To communicate perfectly, to add my thought to your thought perfectly, I'd need an infinite number of descriptors defining cat, sat and mat, which is impossible unless you're God.

Not only is simple addition impossible, so is the simplest communication. Language of any sort always involves a degree of pretence and approximation.

As for Platonic ideals etc, I'm a bear of small brain and could never make much sense of them.

Cheers.

[Leslie]

[robsia]

You know, it's funny how my 4 year old dughter has no trouble with the concept of addition but a grown man seems to find it very difficult to comprehend.

Maybe you should go back to school if you cannot understand the simple statement that two objects need not be identical to add them together.

I don't know how you people live without being able to do Maths. Do you argue with the grocer when you go to buy apples and tell him he can't possibly have more than one, when he has a barrow-ful? How do you follow a recipe?

Besides which, pure maths is not about objects it's about the numerals.

[postodave]

[robsia]

[postodave]

Yes, its me doing two consecutive posts again. There is a thing in Carl Jung's autobiography where he says that at school he could never believe the algebraic expression of the axiom of equals (if a=b and b=c then a=c ie things equal to the same thing are equal to each other) surely he said a=a b=b and so on. He related this to his later rejection of western logic in favour of a more mystical non-dualist approach.

I can see that perhaps there is a sense in which mathematics does not fit the matterial world. Though I think a certain person is exagerating this. But suppose it doesn't. Suppose that I have this clear and distinct idea that 2+2must=4 and no clear examples of this. Then where does my idea come from, what does it refer to. This is Plato's argument in a number of places (somewhere in The Republic for one) He uses geometry rather tham number. I have this idea of a circle, say as a shape where every point on its outside is equally distant from its centre, and I encounter circles, but none of them fit this idea perfectly, so the circle must be an ideal circle to which these others partially correspond. When applied to be number this is exactly Godel's idea. My thought somehow contacts this noetic realm where the facts of mathematics exist as pure ideas timeless, non matterial. And maths works, you know it may just be ideas but you can use it to describe and manipulate the universe. Incidentally on the same tack if no two things in the cosmos are perfectly identical whence comes this idea of identicality.

You can add an apple to an orange, that is easy as someone has pointed out you say two pieces of fruit. But can you add two apples to three concepts. No, so things have to be commensurable in some way to add them together, but this does not mean they have to be identical. But things do not behave exactly like pure numbers. eg my dividing by zero example where the thing you can do with marbles does not give you mathematical truth. Nothing in daily life corresponds to dividing by 0 and getting infinity. And yet we can see why that is the right answer.

[postodave]

[robsia]

I have also heard it is impossible to draw or create a perfect circle. The very materials used to create one create imperfections in the circle.

However, the mathematical concept of a perfect circle is sound, even if one could never exist in actuality (damn it, now I'm beginning to sound like I'm talking gibberish - I hope you understand what I mean).

if a=b and b=c therefore c=a is perfectly accurate.

if a=1/2, b=2/4 and c=4/8 then it works.

In algebra the same number can be denoted by different letters or combinations of letters and numbers. It's called an equation. If it weren't the case there could be so such thing as the lovely: that we all had to learn in school.

[alecto]

This is all fascinating! I wish I could get students to think about all these things when I try to teach mathematics. I also wish I could get teachers to think this way when they teach me mathematics.

In any case, some comments:

First, back to adding up the four kids or the two apples. The issue here is an issue of names. Do you remember the term "polynomial"? It means "many names". When we say "you can add like terms" what we mean is "you can add things with the same name". Whenever we reduce something like two apples to either the phrase "two apples" or some mathematical expression like "2a" we're reducing the complexity of the situation by binning together the things with the same names. Whether we can do this depends only on what we mean by the names. Suppose we have ten fruit: two gala apples, three macintosh apples, and five oranges. If I insist on keeping "macintosh" and "gala", I can't add the apples, but if I decide to use the large category, I can say "five apples". I can then choose fruit as an even larger category and say "ten fruit". So it depends on the categories. Let's say the four kids are two girls and two boys. Once I make that split, I can't add them anymore. If I now say they are Lucy, Edmond, Susan, and Peter, I can't add them at all, because now they are uniquely named.

What 2 apples + 2 apples = 4 apples means is that two items that belong to the category "apple" added to two more items belonging to this category yields four items belonging to the category. The fact that adding or multiplying anything we talk about gets us into this category business is one of the things that separate mathematics from hard reality. Reality consists of fantastically complicated objects that we have to simplify tremendously if we are going to do arithmetic concerning them. One of the reasons the Greeks loved geometry was that they could do proofs like they could with arithmetic yet there was still some of the intrinsic nature of the objects left in. It wasn't just words anymore, it was the real shapes of buildings, objects, etc, at least to some extent.

The language question for math, by the way, is the same one there is for translation that complicates the whole issue of what certain things in the Bible mean. The assignment of objects to categories is not trivial. A simple example is that there are no words in English for the kinds of insects that are listed in the OT as being edible, so the translators of KJV picked four approximate categories, some of which in modern usage include no objects even close to those intended in Hebrew.

And it's not just bugs that defy "metacategorization". "Sin" and "faith" do not pick out the same things in English, exactly, as do their Hebrew or Greek counterparts, meaning that some things we call "sins" or "acts/signs of faith" would not be picked out by Hebrew or Greek speakers as examples of the categories picked out by the Hebrew or Greek terms. Ancient Christians were not so affected by these things as moderns are, because they lived in an openly multilingual culture and knew they were dealing with all these different categories. When someone like Augustine tried to discuss things, he often sounded like a philosopher, because he was always trying to get to the bottom of what things meant. Despite the fact that many of us might not agree with what he came up with, I think he was doing a very good job trying to make sense of what he knew was not a trivial problem.

Now I just ran onto how to figure out things like "faith". Rather than trying to define the word somehow, go read all of the examples of the acts of faith and figure out what the commonality is for yourself, without trying to translate into any language. I don't believe all of the things in the Bible are necessarily true, so I might expect some things to just not "fit in", but my list of acts/attitudes will be less like "x acts of faith" and more like a personalized set of examples. Going back to the example of the four kids, it'll be less like "four kids" and more like "Lucy, Edmond, Susan, and Peter".