It makes no sense to say that there are “more” real numbers than natural numbers. No, neither set has any size so you cannot speak of one being larger!
There are “proofs” that go something like this:
Take the natural numbers from 1 to 3. There are three of them. But, take all the real numbers from 1 to 3, there are more of them! Now take any number of natural numbers from one to X. There are always more real numbers between 1 and X, so therefore there are more real numbers than natural numbers.
This does not work. It does not prove that the “infinite” set of real numbers is larger than the “infinite” set of natural numbers.
All the other proofs share fundamentally the same flaw: Assuming that infinity is something other than a potentiality and then assuming infinite sequences and series have a size.
But, they don’t! Therefore, you cannot use them to prove there are different sizes of infinity.
Infinity does not have a size. Something is infinite, means, it has a certain potentiality. There are no “sizes of potentialities”, therefore there are no sizes of infinity.
Talking about “countable” or “non-countable” infinities does not salvage the claim of sizes of infinity. By countability, they are really referring to mapping between elements of sets, not size or quantity.
Mathematicians do not use “infinity” in the same sense used here, which is, a potentiality to progress in a sequence. Their definition of “infinity” is vague and sometimes incoherent.
That is one example. At least some of the listeners will disagree on this point as well. Perhaps infinity is something we should talk about in a future podcast episode as well and offer a more elaborate argument. But, for now, this will do.
What do you think about the mathematics involved in quantum physics and its interpretation? Where do you think the problem lies in this field?
This is a very problematic area of physics. But, before we go further, we need to draw a distinction here between the mathematics of quantum physics and the physical interpretation of that mathematics.
The mathematics of quantum theory is remarkably successful and its results have enabled a great many advancements in physics, other areas of science and technology. This is the area of quantum physics which has been the most successful. Using this and its predictions has allowed us to gain a very good understanding of many of the relationships between things at the subatomic level and to develop technologies such as the transistor, lasers, atomic clocks and so forth.
The mathematics of quantum theory has been vindicated time and time again. So, at this point, it would be fairly silly to question it. At least to question the more fundamental and important equations, such as the Schrodinger equation and so forth.
But, the problem is in much of the interpretation of the mathematics. While a great many of the predictions made using the mathematics of quantum theory have been verified, that does not justify many of the interpretations of the experimental evidence that have been reached.
Central to quantum theory is the idea that subatomic entities like electrons have a dual identity, that they are both waves and particles at the same time and that not all of their properties are definite.
Now, how does one justify such bizarre claims? They usually point out that their experiments verify that these things are the case. But, the experiments do not, in fact, show this. Nothing can demonstrate that an electron is a wave and a particle at the same time or that it is inflicted with vague properties or that a photon can magically teleport from A to B without crossing the intervening distance.
Yes, the experiments probably do demonstrate something. But, one has to be careful how they are interpreted. One always has to be careful. In fact, the manner in which one interprets an experiment is contingent on their metaphysics, which the interpretations of these experiments illustrate.
If you have a rational metaphysics, you will tend to interpret your experiments in a way that does not allow for contradictions. You will know that if you think you see such things, that either your experiment is flawed or that you have reached a faulty conclusion somewhere.
The problem with quantum physics is that there are a lot of bizarre interpretations of experiments. This is because the founders of quantum theory all started with irrational philosophies and then conducted experiments which seemed to give strange results. Such as the apparent wave-particle duality of light.
Instead of questioning whether or not their experiments actually show that light is both a particle and a wave, they chose to believe that is what the experiments show. Why? Because they have an irrational metaphysics that leads them to expect that reality is full of contradictions and is as Bohr liked to put it, quote, “inflicted with a kind of vagueness” unquote.
That is the problem with quantum physics. While it has undoubtedly had many successes, its explanations of the things it attempts to explain are mystical and or non-physical and thus frequently do not contribute to a proper understanding of reality. In fact, this is worse than offering no explanation.
Now, it should be noted that there is a reason quantum theory is the way it is. It is not because reality is really like that and we have to accept that. It is because that is what the founders of quantum theory expected to find. It is amazing what you can prove when you assume your conclusion and then go out to prove it.
It is widely believed that the strange claims of quantum theory have been proven beyond any reasonable doubt. This is not so. The experiments do not, in fact, demonstrate what most allege that they do. And no experiment ever could. That would require reality to not operate as it does.