16. Modern day philosophers: Mathematicians trying to unravel infinity

 15. Modern day philosophers: Mathematicians trying to unravel infinity

 

Unsurprisingly, this is also the same sort of confusion that arises amongst the modern day philosophers of mathematics when trying to unravel the nature of Infinity. When mathematicians try to make a formula to represent the idea of the unlimited or the concept of ‘infinity’ then it starts to go against our common sense understanding of reality. This is because it is mathematically and scientifically impossible to measure anything infinite in our limited lifespans as any measurement we make that seems very large (and infinite)  could actually mean that the data is just very large (larger than our lifespans) rather than infinite. So for example if a scientist wanted to send a probe/light ray to the end of the universe in order to time its return (and therefore its distance that it has traveled) but does not get a response in his lifetime- could the successive scientist (as the first one has passed away)conclude that the universe is infinite as he still hasn’t received a reply from the edge of the universe? No - for the alternative explanation exists that it is just  further away than the time it takes for a response to reach him (and any successor). It would therefore be scientifically impossible to conclude (and thus speculation) that the universe is infinite because we do not have infinite time to observe. All we can  do and should rationally do is base our thought on the conclusive reality (that the universe  needed a creator to exist) and not the speculative reality (that the universe is infinite).

 When mathematicians  tried to quantify the infinite then they came out with different understandings of what it means mathematically. For instance Cantor, using his set theory demonstrated different  types of infinity such as transfinite, but this led to contradictions in the application of his formula, and for paradoxes to arise. Needless to say even in mathematics there are disagreements and this is because there is no such thing in our reality as infinity to be able to verify any speculative formulae and conclusions.

One such absurdity that arises when an infinite sum of finite things (such as the celestial bodies within the universe) is assumed can be demonstrated as follows. Imagine  an infinite sum of marbles. If we were to halve the marbles then both halves would be equal to infinity. In fact any fraction of the infinite sum of marbles would equal infinity. This then produces an apparent contradiction that the part is equal to the whole. Further if we were to take three marbles out of the infinite sum of marbles then the remaining marbles would still equal to infinity. But the 3 marbles that have been taken out would be a fraction of the overall marbles. Yet this contradicts the principle we established earlier which is that every fraction of the infinite sum of marbles would equal to infinity. Yet the three marbles do not equal infinity. Thus something cannot be infinite and finite at the same time, because of this and many other contradictions it is absolutely clear that the sum of finite things must be finite. (We can also apply this to our previous discussion about the creation of the universe, for those who claim that the universe needs no creator as it is infinite in itself: because the universe is made up of finite bodies within space, and because we can measure parts of the universe which are finite distances then the whole universe is finite even though it be very large indeed!)

Again the problem is that we don’t know what infinite means in our reality and when we try to apply the mathematical formula to life it doesn’t make sense.To illustrate again,  if you imagine a rope stretching to infinity in both directions. It runs to the end of the universe. It is, in essence, infinite. Now look at the space all around it. That also runs to the end of the universe. It is also infinite. Both are infinite, but are they the same? Isn’t one infinity bigger than the other? It doesn’t make sense but Cantor wanted to demonstrate this nonsensical idea mathematically using set theory. This demonstrates the mismatch between abstract mathematical ideas and reality.

Set theory demonstrated here:

The problems here arise in mathematics because it is not based in reality- the numbers that mathematics represents are not real actual tangible and sensed things but are an abstract construct: you can’t hold a ‘2’ in your hand and add it to something that is a ‘3’ to get a ‘5’. What you can do is find 2 items and find another 3 items and put them together and then if you count them you will get 5 items. So the numbers do not exist outside of the reality they represent.

Likewise in the mathematics of the infinity that Cantor delves in- there is  a confusion about reality. In Cantor's set theory he (and mathematicians) presuppose  “all positive integers” (all positive whole numbers) as part of their demonstration of how they can line up with integers from a different class (for example negative whole numbers) which he calls a “set”.  To say, “all the positive integers” is to presuppose an error. Sets aren’t tangible things existing out there in reality. They are constructed by our minds to understand an idea. All sets are exactly as large as they’ve been constructed. Therefore there is no such thing as “all the positive integers”.

 

It’s like asking, “How many words does the largest sentence have in it?” And when you respond, “I don’t know, but at any given time, it’s a finite amount”, they say, “No! I can just add a word to it! It’s an actually-infinite sentence with an infinite number of words!” Just because you can always add another word, doesn’t mean an “actually-infinite sentence” is out there.

 

 Therefore any number or formula that tries to represent infinity is going to be full of speculation and doubt- it is just an abstract idea because in our reality there is no such thing as infinity. It ends up being  just fanciful philosophy as the ancient philosophers would talk about without any real knowledge.

Is it any surprise then that mathematicians who spend their lives trying to prove their fanciful philosophy as real when there is no way to verify it are usually insane or end up insane (as Cantor unfortunately did)– living in a different reality- just modern day Pyrrhos! (as mentioned above Pyrrho was a philosopher who used to deny reality and was always in harm’s way!)

 

Foot note:

There is also another distinction to be considered of the nature of the infinite that is often conflated together as a source of confusion and doubt about God: indefiniteness in number doesn’t equate to unlimitedness in power- can we understand it in our limited exposure to our reality of the limited world?

 Descarte: God alone is strictly infinite. Yet there are created beings, like extension and numbers, which are unlimited in certain respects. So Descartes introduces a technical distinction between the infinite, “that in which no limits of any kind can be found,” and the indefinite, that in which “there is merely some respect in which I do not recognize a limit” . An ordinary body, for example, is limited in size but unlimited in divisibility. The extension of the universe beyond the earth and stars, which Descartes sometimes calls “imaginary space,” is unlimited in size but limited in power, intelligence, and the like .But Descartes also emphasizes an epistemic side to the distinction. Infinite things are those that I “understand” to be absolutely unlimited (in all respects), while indefinite things are those in “which, from some point of view, we are unable to discover a limit.” Extension, for example, is indefinite because “no imaginable extension is so great that we cannot understand the possibility of an even greater one. But I understand God to be actually infinite “so that nothing can be added to his perfection”