I wrote a comment that captures a core part of what I'm trying to explain, so I will copy it here in its own post. > If we take as assumption that everything humans have observed has been made up of smaller physical parts (except possibly for the current elementary...
Introduction Aside from the lowest levels of physics itself, effectively everything that humanity has studied scientifically has been found to reduce to smaller physical parts. This idea is strongly backed by many, many, observations. Because this idea is strongly backed by evidence (which I will not list here), it follows...
Warning: The argument in this post implies bad things about reality, and is likely to be bad for your mental health if it convinces you. If you're not sure if you can handle that, consider skipping this post for now and coming back later. Introduction At my best, when I...
Since multiple answers here mention equality being undefinable in first-order logic, I want to say that that's only true if there are an infinite number of constants/functions/predicates in the language. Since set theory can be formalized using only a single predicate, it is possible to define equality this way:
(Where x and y can be the same variable, but z must be a different variable from them both)
By simply replacing every instance of the formula "x=y" with this definition, set theory can be formalized in first-order logic without equality.
So what part of a mathematical universe do you find distasteful?
the idea that “2” exists as an abstract idea apart from any physical model
It's this one.
Okay, but if actual infinities are allowed, then what defines small in the “made up of small parts”? Like, would tiny ghosts be okay because they’re “small”?
Given that you're asking this question, I still haven't been clear enough. I'll try to explain it one last time. This time I'll talk about Conway's Game of Life and AI. The argument will carry over straightforwardly to physics and humans. (I know that Conway's Game of life is made up of discrete cells, but I won't be using that fact... (read 511 more words →)
My view is compatible with the existence of actual infinities within the physical universe. One potential source of infinity is, as you say, the possibility of infinite subdivision of spacetime. Another is the possibility that spacetime is unboundedly large. I don't have strong opinions one way or another on if these possibilities are true or not.
The assumption is that everything is made up of small physical parts. I do not assume or believe that it's easy to predict the large physical systems from those small physical parts. But I do assume that the behavior of the large physical systems is determined solely from their smaller parts.
The tautology is that any explanation about large-scale behavior that invokes the existence of things other than the small physical parts must be wrong, because those other things cannot have any effect on what happens. Note that this does not mean that we need to describe everything in terms of quantum physics. But it does mean that a proper explanation must only... (read more)
I completely agree that reasoning about worlds that do not exist reaches meaningful conclusions, though my view classifies that as a physical fact (since we produce a description of that nonexistent world inside our brains, and this description is itself physical).
it becomes apparent that if our physical world wasn’t real in a similar sense, literally nothing about anything would change as a result.
It seems to me like if every possible world is equally not real, then expecting a pink elephant to appear next to me after I submit this post seems just as justified as any other expectation, because there are possible worlds where it happens, and ones where it doesn't. But... (read more)
I will refer to this other comment of mine to explain this miscommunication.
Reasoning being real and the thing it reasons about being real are different things.
I do agree with this, but I am very confused about what your position is. In your sibling comment you said this:
Possibly the fact that I perceive the argument about reality of physics as both irrelevant and incorrect (the latter being a point I didn’t bring up) caused this mistake in misperceiving something relevant to it as not relevant to anything.
The existence of physics is a premise in my reasoning, which I justify (but cannot prove) by using the observation that humanity has used this knowledge to accomplish incredible things. But you seem to base your reasoning on very... (read more)
Okay, let's forget the stuff about the "I", you're right that it's not relevant here.
For existence in the sense that physics exists, I don’t see how it’s relevant for reasoning, but I do see how it’s relevant to decision making
Okay, I think my view actually has some interesting things to say about this. Since reasoning takes place in a physical brain, reasoning about things that don't exist can be seen as a form of physical experiment, where your brain builds a description which has properties which we assume the thing that doesn't exist would have if it existed. I will reuse my example from my previous post to explain what I mean... (read more)
I don't say in this post that everything can be deduced from bottom up reasoning.
I wrote a comment that captures a core part of what I'm trying to explain, so I will copy it here in its own post.
... (read 458 more words →)If we take as assumption that everything humans have observed has been made up of smaller physical parts (except possibly for the current elementary particles du jour, but that doesn’t matter for the sake of this argument) and that the macro state is entirely determined by the micro state (regardless of if it’s easy to compute for humans), there is a simple conclusion that follows logically from that.
This conclusion is that nothing extraphysical can have any predictive power above what we can predict from knowledge about physics. This
Aside from the lowest levels of physics itself, effectively everything that humanity has studied scientifically has been found to reduce to smaller physical parts. This idea is strongly backed by many, many, observations. Because this idea is strongly backed by evidence (which I will not list here), it follows that any idea which contradicts it has strong evidence against it.
This means that any theory which is compatible with reductionism and all our other observations ought to be considered much more likely to be true than those which are not. In this post, I will explain how to make sense of mathematics in a world where everything seems to reduce to smaller physical... (read 1217 more words →)
Warning: The argument in this post implies bad things about reality, and is likely to be bad for your mental health if it convinces you. If you're not sure if you can handle that, consider skipping this post for now and coming back later.
At my best, when I used to write, I would write characters first by thinking about their personality, beliefs, thoughts, emotions, and situation. Then I would ask myself "what does this person do?", and let mysterious processes inside my brain automatically come up with the behavior of that character.
So this is how I define a mental model of a person: the collection of processes and information inside a brain... (read 2388 more words →)
That's also true in first-order logic with equality, since nothing except convention stops us from considering models where multiple objects are equal according to the equality predicate. The choice to exclude models which include duplicate objects is just a side-condition used to filter out inconvenient models when studying semantics. We can include such a side-condition when considering the semantics of set theory without equality, too, so it doesn't seem fair to me to single it out as being uniquely incapable of defining equality.
(In fact, I'd argue that this also applies to second-order logic. Second-order logic can be given Henkin semantics, which have all the same idiosyncracies as first-order semantics. Using these semantics,... (read more)