Actually, they are not. The whole idea of an axiom is part of a theory (formal logic) and that theory has no axioms. Even within the framework of formal logic, there are logics with no axioms, only rules of inference.
Sure. Go look at yourself in a mirror. Observe that the thing you see in the mirror looks very similar to other things around you that you call "people" or "humans". Observe further that the thing in the mirror behaves differently from other humans. You can control the behavior of the thing in the mirror in ways that you cannot control the behavior of other humans. Observe that when you try to touch the thing in the mirror it feels different than when you touch one of these other humans. It will feel cold, smooth, and hard, like glass, not soft and warm like flesh.
Now, go find another human and ask them to look in the mirror with you. Observe that you can now see a second human-looking thing in the mirror, but this second human-looking thing has yet another set of properties. For starters, this second human-looking-thing in the mirror doesn't look like you, it looks like the other human whose assistance you have solicited. If you try to touch the second human-looking-thing-in-the-mirror you will find that you will not be able to. The first human-looking-thing, the one whose behavior you have direct control over, will thwart your efforts and get in your way. You will find that the control you exercise over the first human-looking-thing-in-the-mirror is constrained in a way that makes it impossible to touch anything in the mirror other than it.
Now ask the other human whose help your have solicited what they see. You will find that their perceptions are similar to yours. They will (almost certainly) tell you that they also see two human-looking-things in the mirror, and that one of them, the first one, looks and acts just like you. You will find that your assistant will agree with you on many other observations as well. They will agree on the location of the mirror, the fact that it feels smooth and hard. If there are other objects nearby they will agree on the location and appearance of those objects.
Now turn out the lights. Observe that you can no longer see the mirror or any of the objects in it that you could see before.
From this and a whole lot of other similar observations you can construct the following theory: you are a thing that exists in space which has three dimensions. In this space there is this stuff called "light" which moves in straight lines (more or less) from sources and ultimately enters part of the space that you occupy that you call "eyes". When this happens, you obtain information about other things that occupy the same three-dimensional space that you do. Some of these things are very similar to you. They also have eyes and other body parts, all of which are things that exist in the same 3-D space and occupy particular subsets of that space. The particular subset of the space occupied by things changes, and this is an indication of the existence of something called "time", which is a radically different kind of thing from the kind of thing you are and the kind of thing light it. You and light exist in space. Time does not. Time is some seriously weird shit.
Do you need me to elaborate further, or are you getting the idea?
No, because I feel like there are a lot of background assumptions involved in that process, many of which are either unexamined or necessary for the theory-making process. These background assumptions are reinforced every time the theory allows me to make correct predictions and challenged every time I make incorrect predictions. Incorrect predictions allow me to update the theory or the assumptions.
When scientists do the a formal version of the same process as you have described, they make similar background assumptions. Perhaps you can explain how this whole process transpires without assuming anything?
In your informal example, I’m assuming that the person I’m speaking to is an honest interlocutor. In the scientist’s formal counterpart, he is assuming that the phenomena he observes correspond to the representation he assigns in the formal model. For a specific example, in order to infer the composition of a star, one must assume that the spectra emitted by a star are products of the same emission process as observed in a laboratory here on Earth.
> I’m assuming that the person I’m speaking to is an honest interlocutor.
OK, but you have taken it upon yourself to assume that. I never asked you to assume it. It is entirely possible that the person you are talking to is not being honest. They could be gaslighting you. It is even possible that they are not a person. They could be a shape-shifting space alien disguising themselves as a person as part of a grand conspiracy to prevent you from discovering the fact that you are a 13-dimensional being with super-powers who would take over the universe if you became aware of your true nature.
So... among these two theories which I have now proposed to explain your observations, which do you think is more likely?
I’m not sure which is more likely, I have interacted with honest interlocutors and with gaslighters before. That’s kind of the point, isn’t it? I can easily assume good faith and move forward, which involves an assumption of good faith. Or I can refuse to assume anything and do what now?
Seriously? You assign the same probability to classical physics as you do to people around you being shape-shifting space aliens disguising themselves as people as part of a grand conspiracy to prevent you from discovering the fact that you are a 13-dimensional being with super-powers who would take over the universe if you became aware of your true nature?
You'll have to excuse me if I find that hard to believe.
No, the 13th dimensional being was your theory. I just didn’t believe the person in the scenario you described, the same way you are doubting my sincerity in this little conversation.
> I just didn’t believe the person in the scenario you described,
Why not?
> the same way you are doubting my sincerity in this little conversation.
Not quite the same way, because I can answer the "why not" question, and I'm pretty sure you can't.
But let's leave that aside and try a different experiment. I presume you have a mobile phone? If not, substitute some other common object. For the next week, for everyone you encounter, show them your cell phone (or whatever common object you choose) and ask them, "Do you see this? What is it?" and see if you don't observe a certain consistency in their answers. How do you account for that consistency?
Because thats the exercise we agreed to practice; how to make a theory without assuming anything. I neither believe nor disbelieve this person, how am I to use their testimony to construct a theory without taking it as some English words with particular meanings that are presumed to be presented in good faith as an accurate representation of their perspective?
> Not quite the same way, because I can answer the "why not" question, and I'm pretty sure you can't.
Ok how do you answer the “why not?” question?
> How do you account for that consistency?
Ok if I correctly understand you, you’re saying that I can infer a consistency between phenomena by observing that they appear to be similar. Is that an accurate restatement of your thesis?
> I neither believe nor disbelieve this person, how am I to use their testimony to construct a theory without taking it as some English words with particular meanings that are presumed to be presented in good faith as an accurate representation of their perspective?
By constructing all possible explanations you can think of for the totality of what you observe and choosing the one that seems most likely to be correct. Possible explanations include:
1. The world is more or less as it appears to be. Most words mean more or less what you think they mean most of the time. Most people usually give truthful answers to mundane questions. And so on.
2. The world is not as it appears to be, and the fact that it appears different from its true nature is all part of a grand conspiracy to conceal the truth from you.
3. The world is not as it appears to be, but the fact that it appears different from its true nature is not a conspiracy, it's just that you happen to inhabit a particularly weird branch of the multiverse where things happen to appear as they do against astronomical odds.
4. You are actually a butterfly dreaming he is a Chinese philosopher dreaming he has hanging out on Hacker News.
5. Shrimp noodles every second purple kilogram.
6. Bzork snoz biff boff.
Constructing additional possibilities is left as an exercise.
> Ok how do you answer the “why not?” question?
Because I don't believe that you actually give any credence to the theory that people around you are shape-shifting space aliens disguising themselves as people as part of a grand conspiracy to prevent you from discovering the fact that you are a 13-dimensional being with super-powers who would take over the universe if you became aware of your true nature. In fact, I would give long odds that that possibility never entered your mind before I suggested it, and that you consider the idea to be completely absurd.
> you’re saying that I can infer a consistency between phenomena by observing that they appear to be similar.
Not quite, it is the appearance of similarity itself that I am asking you to account for.
In general I am asking you to account for the patterns and regularities in your perceptions. I am asking you to account for the fact that you appear to be able to navigate your existence, to obtain enough food and shelter to survive, to avoid falling off cliffs and being hit by buses. I am asking you to account for the fact that you are engaged in this exchange, that you are interacting with some kind of device (or at least perceiving yourself to be doing so), the fact that the text you are reading right now make more sense to you than #6 above, and that I was able to predict that this would be the case.
https://en.wikipedia.org/wiki/Hilbert_system