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In Meditation III, Descartes argues that we cannot gained the idea of an infinite God through reflection or perception – the idea must be innate. We are finite beings and incapable of creating the idea of an infinite being. An infinite being is more real than us. But the Causal
Adequacy Principle states that a cause must be at least as real as its effect. Let us look more closely at the finite and the infinite.
A finite collection of things is one that can be put into a sequence with a first and last member. We say that there are finitely many things. We can give a particular number to the collection. So, there are seven books on my desk, for example. We can get the idea of ever bigger numbers by adding numbers together: 74743, 2394284, 4532 9323492 and so on. But the infinite is not a really big number. It is something else entirely. To say that there are infinitely many things is to say that the sequence has no end. There is no end to the sequence of natural or prime numbers, for example:
 1, 2, 3, 4, 5,….
 2, 3, 5, 7, 11,…
Now, Descartes says that we don’t understand infinity by thinking not finite. In Meditation III, he says that we only understand we are imperfect – finite – beings by seeing that we lack powers and hence that we are not infinite. We get the idea of finite from the infinite, not the other way around.
Furthermore, to think simply not finite is to not distinguish between an actual and a potential infinity. The rough idea is this. An actually infinite collection has infinitely many things in it all at the same time. A potentially infinite collection is one that could be indefinitely increased.
An image might help. Imagine you are walking down a road. A potentially infinite road is one that could always be extended without ever hitting a limit. An actually infinite road is one that is already built and infinitely long.
Now, it is important to Descartes that our idea of the infinite is of an actual infinite. He accepts that we can invent the idea of a pseudoGod as potentially infinite. Here’s the recipe.
Consider yourself. You have some degree of knowledge, power, goodness, and so on. You are also aware you could have more knowledge, power, goodness, and so on. So, with a leap of logic, you can imagine a being whose knowledge, power, goodness and so on increases indefinitely. Such a being is potentially infinite – increasing without limit. But God is and never was not a work in progress. God is and always was complete – actually infinitely knowledgeable, powerful, good, etc.
The difference between actual and potential infinity goes back to Aristotle. He used it to solve the motion paradoxes of Zeno. As you may recall, to walk 100m, I must first walk half of it:
50m. To walk that 50m, I must walk half of it: 25m. And so on. Suppose the space is infinitely divisible into the following sequence of distances…
 50m + 25m+ 12.5m + 6.25m +…. +….
Suppose, for example, I walk at 1 m/s. The time it will take me will be:
 50s + 25s + 12.5s + 6.25s + … + ….
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Since to cross any distance takes time, it will take me an infinite amount of time to cross the infinitely many distances. If velocity is distance over time, then my velocity will be zero.
Motion is impossible! In general, the thinking is that if you have an infinite sequence of finite values, no matter how small, then their sum will be infinity. Infinitely many little things still add up to infinity.
Aristotle argued that it is not as if the 100m is actually pre-divided into infinitely many distances; it is potentially divisible. This was a typically clever response but one that doesn’t really work. If I can infinitely divide the space, then those divisions are in some sense there waiting to be opened up. It took until the development of the calculus in the 1660s before people finally had the mathematical tools to explain how when you sum an infinite number of finite amounts, as above, it doesn’t
equal an infinite amount but a finite amount.
A vital step in Descartes’ argument is that we couldn’t have created the idea of God because we can’t create the idea of the infinite. But this is problematic. For if we really understand the idea of the infinite – really get our mind around it – couldn’t it be a product of our minds after all? And so, by the same reasoning, couldn’t we have come up with the idea of God?
What about the following way out? Our idea of the infinite and of God is not complete but partial or perhaps a bit hazy. Well, if so, then we open ourselves up to a different version of the same reply. A picture that looks hazy may be hazy as it is a bad representation of a clear reality. Alternatively, it may look hazy because reality is hazy. Think of a painting of something in the fog. Now, if my idea of God is hazy, perhaps that’s because God himself is hazy. Or, to put it differently, I have an clear idea of a hazy being. And, once again, if I have a clear idea, couldn’t I be its author?
So, in sum, whether I have an idea of a clear or hazy God, the fact that I have the idea shows that I could be its author and hence that there is no reason to think that there’s a God out there.
Descartes has a way out. He says that we can know something without understanding it. To know something is to have some comprehension of it whereas to understand something is to have a complete comprehension of it. For example, we can know God but not understanding.
Descartes writes:
“I say that I know [God is the author of everything], not that I conceive or grasp it; because it is possible to know (connaître, savoir) that God is infinite and all powerful although our soul, being finite, cannot grasp or conceive (comprendre, concevoir) him. In the same way we can touch a mountain with our hands but we cannot put our arms around it as we could put them around a tree or something else not too large for them. To grasp something is to embrace it in one’s thought; to know something, it is sufficient to touch it with one’s thought.” [27 th May 1630 letter to Mersenne; CSMK
25.]
“Je dis que je le sais [que Dieu est l’auteur des vérités éternelles], et non pas que je le conçois ni que je le comprends ; car on peut savoir que Dieu est infini et toutpuis sant, encore que notre âme étant finie ne le puisse comprendre ni concevoir : de même que nous pouvons bien toucher avec les mains une montagne, mais non l’embrasser comme nous ferons un arbre, ou quelque chose que ce soit, qui n’excédât point la grandeur de nos bras : car comprendre c’est embrasser de la pensée, mais pour savoir une chose il suffit de la toucher de la pensée”.
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We know God and infinitely enough to understand them but not so much as to be the author of these ideas. One may feel that this is a rather sneaky way out!
The mathematics of infinity really came of age in the 19 th century. In the 1870s-1880s, a mathematician called Cantor proved that there wasn’t one infinity but infinitely many infinities.
It’s not possible here to go into details, but it is relatively easy to show two different sizes of
1


2 infinity.
There are infinitely many natural numbers (denoted by N ):
 1, 2, 3, 4, 5….
There are just as many even numbers. This seems counter-intuitive. Surely there are half as many – all the even numbers and all the odd numbers make the even numbers. No. We can
‘pair off’ all the natural numbers with the even numbers:
2


4
3


6
4


8
5


6


7


8


10 12 14 16
…
…
…
…and we’ll never run out. We can do the same with the prime numbers, the square numbers
(1 2 , 2 2 , 3 2 , ….) and indeed any power sequence (1 n , 2 n , 3 n , ….) We can even pair off the real numbers and the fractions or rationals ( Q ) = {0, 0/1, 1/1, -1/1, 1/-1, -1/-1, 0/2,...}. It is easy to lay out a table as before, but it is even easier to see if we consider the following twodimensional layout of the rationals instead:
... ... ... ... ... ...
... 3/-3


3/-2

3/-1

3/1

3/2

3/3 ...
... 2/-3 2/-2

2/-1

2/1

2/2

2/3 ...

... 1/-3

... 0/-3

 
1/-2 1/-1

1/1

1/2 1/3 ...
   
0/-2 0/-1 0/1

0/2 0/3 ...
  
... -1/-3 -

1/-2 -1/-1

-1/1

-1/2

-1/3 ...

-2/-2

-2/-1

-2/1

-2/2

-2/3

etc. ... -2/-3

... -3/-3 -

3/-2

-3/-1

-3/1

-3/2

-3/3 ...
... ... ... ... ... ...
The idea is to start at 0/1 (in bold) and move in a spiral: 0/1, 0/2, 1/2, 1/1, 1/-1, 0/-1, -1/-1, -
1/1, -1/2, -1/3, etc. This spiral line can be considered as stretched out with each rational on it corresponding to a member of N on the ‘straight line’: 0, 1, 2, 3,… This is by far the most paradoxical-seeming result, because it is also possible to prove that that there are as many rationals between 0 and 1 (exclusive) as there are natural numbers:
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The set of rationals between 0 and 1 (exclusive):
1/2

1/3

2/3

1/4

2/4

3/4

1/5

2/5

3/5

4/5

1/6 ...
So there are as many rationals between 0 and 1 as there are natural numbers - and, since there as many natural numbers as rationals (all rationals, the set Q ), there are as many rationals between 0 and 1 as there are in total.
But consider now the collection of real numbers ( R ). A real number is one that can be positive or negative, a whole number, a fraction or a decimal that goes on forever and can’t be represented as a fraction (such as

). Intuitively, there are fewer natural numbers than real numbers, as the real numbers contain the natural numbers and lots of other numbers.
But intuitively, there are fewer even numbers than natural numbers, and yet we now know that that is not true. It turns out, however, that the infinite collection R really is bigger than the infinite collection N .
Let’s try pairing off members of N and R . Here’s how it might look:
N
1
2
3
4
…
R
1.3459343453…
3.3453462786…
20.0000000000…
234.2368992574…
…
Now let’s generate the number k in the following way. k starts with 1 and then the decimal point. The first number beyond the decimal point is generated in the following way:take the first number past the decimal point in the first row and subtract it from 9. In row 1, that first number is 3. Subtract it from 9 and you have 6. So k=1.6 – so far.
The second decimal digit of k is calculated in the same way. Take the second number after the decimal in the second row, subtract it from 9 and put it in. The third, fourth, fifth decimal numbers are calculated in the same way – and so on for all the rest. Using the table above, k starts: 1.6591
Now ask the question: is k somewhere in the list R of all real numbers? If not, we have found a real number without a natural number pair, and so R is “bigger” than N. We’ll now prove this.
Suppose it is some where in R – let’s say at row n. Now we know that the nth decimal of k is
9 minus the nth decimal of the number in the nth row. So we can say that the nth decimal of k is 9 minus the nth decimal of k: call that nth decimal number – which must be
0,1,2,3,4,5,7,8 or 9 – the number z. So:

z = 9 – z
But the only solution to this is z = 4½. So k can’t be anywhere in the table. So there are more real numbers than natural numbers. One infinity is bigger than another.
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