What can we Know?

This might not seem like a very hard question at all – I mean, you know that you are reading this post, you know where you are, and so on. And yet, the field of epistemology is huge, and no consensus has been reached yet. So, today we will try and begin to make an answer to the question, what exactly can we know?

This post has been spun off a recent argument that has been going on for a long time in our Discord server. Why don’t you come join us if you aren’t yet to get in on the conversation? Around June this year, Veritasium released a video called "Maths Has A Fatal Flaw", and it really got polymath talking – so much so the argument is still going on, here I attempt to deal with this.

The Veritasium video covers Godel’s incompleteness theorem. In it, Veritasium does an ok job, but the problems begin soon after. Veritasium does not stress this point quite enough: Godel’s incompleteness theorem does not mean maths is fake, made up, meaningless or simply a convention. This is not really the case at all. All Godel’s theorem says is the only way to prove something in general is to prove it. Huh? Well, simply, imagine a mathematical statement, maybe something like "this computer program will terminate". That statement could be true or false, and in general, the only way to find out is to run the computer program. The proof for something… is the proof. It’s a kind of information problem – how can we know something before we know something.

Anyway, playing off these sorts of ideas, I made, what I thought was a simple observation – all As are As. What the hell does this mean? Well, put simply it means the following "things that are themselves are themselves". To me, this seemed like a very obvious statement, after all, by what means could one deny this to be the case? Dear reader, I had no idea what was waiting for me…

Well the claim was met with resistance. My claim was simple, if all As are As, then all not As are not As (from here on, I will use ¬ to represent not in logic formats). So, to give an example, I could say something like "everything in the universe is either a forklift or not a forklift". This is an example of something that can be known without checking. I trust that you can clearly see that you don’t need to go out and observe every single thing to know that this sentence is true, and heck, you don’t even need to know what a forklift is! However, I was dumbfounded to find that this simple idea was rejected.

The argument went something like this. My assertion that everything is either A or ¬A is only my conception of logic – someone else might not have a conception of "not" or have a system of logic that allows for three states, A, ¬A and… something else? I find this to be totally nonsensical. What other possible states are there? I never thought anyone could think that logic like this was mere happenstance of biology or a simple semantic.

So, the debate turns to mathematics, and questioning if it tells us anything about the real world. I, of course, maintain that it does. After all, maths is an abstract form of logic that this universe inherits from. The planets orbited the sun before humans existed, atoms vibrated before humans existed, and 1=1 before humans existed. According to my opponent, this was not so. Mathematics is merely a "thing humans do", it is simply a "coincidence" that we can use it to make things (air planes, computers and so on). By this point my jaw had hit the metaphorical floor. Is it not clear that mathematics is not the symbols that we use, but the actual concepts behind them? It seems to me that things like Pythagorean’s theorem is absolutely true in the abstract, and yet no triangle has ever existed that has fulfilled it.

This brings me neatly along to my next point. You may be wondering "well, how can we say Pythagorean’s theorem is true if no triangle has ever obeyed it?". This is a great question, but easily answered. The Pythagorean’s theorem, and maths in the general, deal with abstract truths. We do not require a triangle to measure to see that a^2 + b^ = c^2. Of course, any triangle in the real world can never be exactly a right angle. Maybe it’s 90.1°, 90.0001°, 90.0000000000… you get the point. This can not be known exactly. This comes to my main point – rational truths can be known exactly, empirical truths can only be approximated.

Now, we can cover the main arguments of my opposition, the logical positivism, following in the style of someone like Karl Popper. The logical positivists reject the notion of rational knowledge, and instead say there are only two types of statements, hypothetical statements and definitional statements. Hypothetical statements are statements that we can test and know if they are true, scientific statements, so to speak, statements like "this pen is blue" and so on. Definitional statements involve us pointing at things and naming them – "that is a tree" and so on. Now, you might be swayed by this, but this is really quite obviously false. As a quick dent to logical positivism, we can simply apply logical positivism to itself. If there are only two types of statements, which type is the statement "there are only two types of statements, hypothetical statements and definitional statements". Now, either it is a hypothetical statement, and if it is, I have no reason to listen to you since you have not tested it everywhere, and I can always come up with a list of excuses, so to speak, as to why it might be wrong ("but what if you wore this hat and tested it?" and so on). Or, it is a definitional statement, in which case, why should I listen to you? After all definitions are mere semantics. There is of course, only one way out of this predicament, but the logical positivists can not admit it is true; a third kind of statement that is a priori true.

Let us now consider some statements to give some concrete examples, as this is quite confusing. We will begin with testable, hypothetical statements, and I agree, these are hypothetical statements.

What you will notice is that all four of these statements can be reversed or changed fundamentally in some way, and still be hypothetically true:

How would we decide which version of the two statements is the (approximately) correct one? Well, we go out into the world and make observations, tests, measurements and so on. This is empirical, hypothetical, scientific and tells us something about this universe.

Now, let us consider a second set of statements

What do we notice about this set of statements? Let us invert them:

I do not suppose I need to explain by now why this set of statements simply can not be inverted and still be true.

Steven fry once spoke about a similar topic. He took a logical positivist position, and says that it does not make sense to speak of 1+1=2 in the abstract sense. It only makes sense to say "One thing that is empirically identifiable plus another thing that is empirically identifiable is two things that are both empirically identifiable". The issue with his take is that it rests on the bedrock of rationalism – the abstract ability to sum things together. All empiricism is like this, after all, we can never make a conclusion without some rational axioms. If we were pure empiricists, then all we can say is "one day I did this test and got this result, and the next day I did this test and got this result" and never go anywhere from there.

I think I have presented enough for one day, I will leave this here for now. Next time I come back to this topic I hope to discuss the paper "Is Logic Empirical" by Hilary Putnam. Thanks for reading.