I have used the term "to know" unconditionally only once in my comment above, as in We know quite a bit about the natural numbers", In the context of the natural numbers, the term "to know" is actually appropriate and in accordance with the JTB account of knowledge: a justified true belief. The reason why the term "to know" is acceptable in the context of the natural numbers, is because we actually know quite a few theorems that are provable from arithmetic theory and that are therefore true in the natural numbers. Concerning the universe, however, I have used the term "to know" strictly conditionally.LuckyR wrote: ↑October 1st, 2022, 1:42 amSubstituting "suspect" for your use of "know" is more accurate and makes our postings consistent with one another.heracleitos wrote: ↑September 30th, 2022, 10:21 pmWe know quite a bit about the natural numbers even though it is impossible to obverse them all. There is an infinite number of them, potentially or actually, depending on whether you accept the axiom of infinity.
However, it is not because the natural numbers are infinite that there is nothing beyond them.
According to CH (Continuum Hypothesis), there is gap after the natural numbers. Next, there is an entire copy of the real numbers. Next, there is another massive gap. That goes on forever.
If the elusive theory of the universe contains a copy of the relevant fragment of arithmetic theory (Robinson's Q), then the universe has the same structure.
By the way, in a similar fashion, you do not need to observe the entire globe to seriously suspect that it is round. There are in fact numerous, subtle hints that give it away:
Similarly, you can detect the structure of the universe from subtle hints. One such subtle hint is the general self-referential lemma:7 Ways to Prove the Earth Is Round (Without Launching a Satellite)
The first person to estimate the circumference of the Earth was a Greek mathematician named Eratosthenes, who was born in 276 B.C. He did so by comparing shadows case on the day of the summer solstice ... On a flat Earth, there wouldn't have been any difference between the length of the shadows at all.
For every property about logic propositions, there exists a true proposition that has the property, or, a false proposition that doesn't have it.
If this lemma is true inside our universe, then we know what its structure must be.
By the way, Hubble's red shift is also just a very subtle hint, which tells us that the universe must be expanding.
By the way, the JTB account of knowledge is perfectly compatible with the theorem of soundness: If a proposition is provable (=justified) from its theory, then it is true in all its interpretations (=universes). Therefore, the JTB account of knowledge is the canonical definition for mathematical knowledge in all systems in which soundness is provable.