A little digress into math.
Fact: There is no real infinity but God.
Objection: Your math and physics problems deal with infinities. How do you reconcile it with the fact that there is no real infinity but God ?
Answer: Representation is different from what is real. To give an example – in real world, two particles can attract, protons and electrons for instance. While representing on paper, one is assigned (+) and other (-). That does not mean negative, as a quantity, exists. Negative is only a representation for an opposite behavior. Likewise for infinity. If we can imagine something to continue endlessly, we represent that continuity as infinity. Thus there are infinite series, infinite numbers, infinite time … etc.
But pause here for a moment – is there any quantity in our universe that is actually infinite ?
The mathematical meaning of infinity is not the same as a very big number. Big numbers are still finite and can be broken down. The amount of sand particles on earth, the count of water drops on all the earth, the count of atoms in all the universe, though are very large numbers, fades to nothing, against infinity. Until now, no man has ever discovered anything in this universe, compared to which, both big or small, fade to nothing. We could be a tiny speck compared to the universe, but we are still a tiny speck, not zero.
Large but not infinite
A natural property of infinity is that one cannot measure even a small portion of infinity because that itself would be an infinity. In mathematical terms, that’s the same as saying “infinity divided by a finite number is the same as infinite”. What would be the corollary ? If there is a quantity we are capable of measuring, there cannot be infinite amount of that quantity. For instance, we, as humans, are capable of measuring a very small amount of sand with our hands. Being a finite count, that small amount of sand cannot be part of an “infinite sand dune”, but a big “finite sand dune”. Likewise we can measure the time elapsed between two events. Though we can think of time running endlessly, time cannot be infinite simply because we can measure a finite portion of time. Do you know of anything that you cannot measure even a small part ? If not, it is only rational to conclude that infinity, while represented by the lemniscate symbol – ∞ – on paper, cannot be observed, felt, or understood by our senses.
I will answer two more objections here.
1) If I get infinity as an answer to a practical physics problem … does that mean I’m wrong ?
Based on the problem, and on a strictly practical sense, yes. Here is a good example:
http://mathforum.org/dr.math/faq/faq.fly.trains.html
The problem is that two trains approach each other, and a fly buzzes from the nose of first train to the second, and then from the second to the first and so forth. The problem asks to find how many trips the fly makes before the two trains collide. While I’m not solving the problem here, the answer your professor expects is infinity. It’s worthwhile to see how infinity is obtained, from the article link.
If math says infinity is the answer, why do I say it’s wrong ?
Just try to think how that could be possible. Trains would indeed meet at a finite time, and the hypothetically tireless fly has actualized an infinity when the two trains meet ! No human – not even your professor – would have any idea how that could be possible – to contain infinity in a finite space and time.
So then why does the math say it’s infinity, when it cannot occur. Here is why: The solution does not distinguish real objects from mathematical objects. When solving a problem, you actually translate the real world objects – trains, fly, flying path – into mathematical objects – points and line. The difference is this: trains approaching each other meet, but two points approaching each other never meet. If two points never meet, then the fly, another point, can definitely make infinite round-trips.
“Really? But if I put two dots on my paper, I can well conceive the idea of those two dots moving toward the other and also meeting !”, you might ask.
Well, two dots do meet, but dot is only a representation of a point. In math, a point is not defined, but it’s characterized as infinitely small. Think of two dots on your paper approaching each other and growing smaller and smaller as they approach. “Infinitely small dot” means that those dots can grow smaller and smaller without end. If the size diminishes by half, the size of the dot diminishes in the pattern of 1, 1/2, 1/4, 1/8, 1/16, …. , and continues like that for ever. Each time the dots grow small, it has to traverse some more distance to meet the other. Thus if the size of those dots can diminish infinite times, they will never meet. The fly, another point, can therefore have infinite round-trips. But all of that happens only with lines and points. In reality, trains meet, objects meet, and infinity is never actualized. Even a hypothetically tireless fly, after a few round trips, will come to a halt if it’s jammed between two trains.
As an aside, it’s my view that students ought to be taught such questions so that they understand the limits of theoretical models. Many scientists today think infinity is real if they could obtain solution to a problem by introducing infinity, the infinite oscillation of universe, for instance. It only means you have reached the limit of your model or that you better heed your high school math teacher’s advice – check what went wrong if you get a division by zero.
2) Another objection – Space and time are definitely infinite, if not, what exists outside space and time ?
This, in my view, is the only sound objection. Human mind cannot fathom anything outside the three dimensions of space and a continuously running time. It’s therefore natural and reasonable to assume the space in universe to be infinite, and that time would run continuously to infinity. However such a view is incompatible with both theory and observational evidences made in the last century. Beginning with the Hubble’s law and the much recent detection of gravitational waves from the NASA-developed detector on the BICEP2 telescope, confirm an expanding space in our universe. Expansion of space would imply a finite space, because an “infinite space” cannot expand !
Many prominent scientists, including Albert Einstein, had believed in an infinite universe. However today, the evidence and theory to support it is terribly poor compared to what we have today to believe in an expanding universe with a definite starting point. Given the evidences from observation and theory, and the fact that our mind cannot fathom beyond time and the three dimensions of space, what is rational and wise would be to accept the limits of human mind, than to study pure speculative models for universe’s origins to quench your curiosity for what lies beyond the observable universe.
Note that the adjective “observable” for universe, is a term from the wisdom of ancient religions, which classified the universe as visible and invisible. The Fathers of Nicene wrote the Christian Creed as : “We believe in One God, creator of the visible and invisible universe”. Our empirical sciences deal with the former, not latter. The unobservable or the invisible universe is beyond our intellectual senses.
xavierabraham 