2 + 2 = 4 is a mathematical statement

The relativistic velocity formula that you gave is a physics statement. Anyway the internal structure of the formula still follows the rules of math where 2 + 2 = 4.

I think Crazy Diamond was illustrating the point that “2+2=4” doesn’t say anything about the universe itself but is a useful language game tool in certain arenas.

I think Crazy Diamond was illustrating the point that “2+2=4” doesn’t say anything about the universe itself but is a useful language game tool in certain arenas — Noah Te Stroete

Why?

As you can see the math rules still apply inside the relativistic speed calculation.

I think the relativistic speed calculation is an error in physics rather than math.

I think math is a tool to describe reality just as physics is. Math isn’t inherent in reality. It’s inherent to our minds in describing reality.

Never mind that minds may be inherent or at least inevitable in at least this universe. That’s another discussion, though.

So, what about the real universe? Does this mathematical game, in which (1+1) + (1+1) = (1+1+1+1), accurately describe the cosmos that we live in? — Crazy Diamond

I'd argue that maths transcends and therefore governs the universe. You can write:

2+2=5

Subtract 4 from each side give:

0=1

That implies false=true. That implies logic is broken and there is also no information in the universe (IE bits of information can take on only a single value).

So it seems basic maths and logic must hold in all possible universes. Geometry is a different matter though. Various geometries have additional axioms beyond basic arithmetic/logic that are questionable and may not hold.

2+2=5

Subtract 4 from each side give:

0=1

That implies false=true. — Devans99

But you didn’t start with a true statement. “2+2=4” is true given the conventional meanings of the terms used. “2+2=5” is false given the conventional meanings of the terms used. The operative word is “convention”.

I believe that mind and matter are both necessary components of reality. Math is inherent in minds but not matter. Math is used by minds to describe how the physical world appears to minds, but matter would be quite different without minds.

The title of the OP is 'Is 2 + 2 = 4 universally true?'. So I've assumed it was false to see where it leads and 2+2<>4 always leads to a contradiction. There is no way to make 2+2<>4 consistent so it must always be false.

As I said above, a universe in which 2+2<>4 is a universe with no information in it. IE no matter/energy. A very boring sort of universe.

Maths is just an extension of logic. Maths/logic existed before any minds and before the universe was created. The universe must obey the rules of math IMO.

Mathematics is an abstracted way of thinking about relations, with some basis in external-world relations as we observe and think about them (which doesn't imply that any mathematics is identical to external-world relations, of course), but the bulk of it is extrapolated from that, creating a sort of construction/game upon that in an erector-set manner.

Because of that, there's no reason to say that any mathematical statement is universal.

As it is, no mathematical statement is universally constructed by humans, but we have very stringent socialization procedures in place to enforce conformity to the norms.

I think math is a tool to describe reality just as physics is. Math isn’t inherent in reality. It’s inherent to our minds in describing reality. — Noah Te Stroete

An interesting remark. I guess math began as an abstraction of reality. We started with the basics which meant math was just counting.

The odd/interesting thing is that when we developed and amplified the abstraction and applied it back to the real world something clicked.

I'm not a mathematician so you'll have to take this with more than a pinch of salt but all known laws of science are mathematical. Now if math isn't something inherent in reality we should be seeing some exceptions which I presume we haven't.

It's like someone who sees a theme in a storyline and discovers that all other stories have the same theme. Is it coincidence? Is it God? Is our world mathematical?

What do you think?

all known laws of science are mathematical. — TheMadFool

It's just that that's the language we're using to describe phenomena.

If we used a "natural language" instead, we could say, "All known laws of science are in English."

It's just like the world being in "black & white" with black and white film, or the world being only shades of red/pink/white if we only have red and white paint.

I think math is a tool to describe reality just as physics is. — Noah Te Stroete

actually all physics is, at its core, is a mathematical model of some reality or observation. Which then allows you to change the variables and make predictions. Then test those predictions experimentally to see if the math is actually predictive. If so, great you may have a workable model, if no - can it and try another one.

Now if math isn't something inherent in reality we should be seeing some exceptions which I presume we haven't.

It's like someone who sees a theme in a storyline and discovers that all other stories have the same theme. Is it coincidence? Is it God?

What do you think? — TheMadFool

There is math that doesn’t refer to anything in reality. For example, if String theory is correct, then the fundamental units have a minimum size which isn’t zero. This would imply that zero-dimension points don’t actually refer to anything? Then again, the entire theory and it’s mathematics could very well be pure fiction.

I believe our minds are mathematical, and our minds being a part of the natural world, then I guess in a sense the world is mathematical. Theories and language describe our perception of the universe, however. If the universe couldn’t sustain life, then I think mathematics wouldn’t exist.

A mathematician friend of mine did his dissertation on something that was over my head. I asked him if it referred to anything in the physical universe, and he said he didn’t know.

Isn't 2+2=4 just an instance of the law of identity, A=A? Any something is itself. How could it be that anything in actual physical reality could be not itself?

"2+2" is just another way of saying "4". "2+2=4" is just another way of saying "4 is 4". "This rock is this rock" is an instance of A=A or "this is this". Any time you put two expressions on opposite sides of an equals sign, you are declaring identity. If they are not equal, your equation is simply false.

As for physics, aren't conservation laws just the same sort of thing? This much stuff is always this much stuff. It can't be not equal to itself. No matter what you do, you can't make it not equal to itself.

If you ever discover that what you had put on one side of an equals sign is in fact not equal to what you had put on the other side, this simply indicates a prior failure to understand what you are dealing with, maybe what the basic stuff being conserved in fact is. If something like mass-energy equivalence seems like it is a statement that something is not itself, this just means that you fail to understand what mass and energy are at a more fundamental level.

Einstein's theories don't involve violations of basic conservation laws or the law of identity or the law of non-contradiction. If they were to involve such violations, they'd be wrong.

With regard to speed, isn't it just the case that Einstein gave us a deeper understanding of what it is that is being conserved? Regardless of the frame of reference, the spacetime interval is invariant, no?

If you show that a previously believed equation doesn't hold, you haven't demonstrated an actual instance of A in fact equaling non-A or 2+2 equaling 5. Rather, you are showing that what you had put on one side of the equals sign is in fact not the same as what you put on the other.

With regard to relativity, a difference in the frame of reference is what makes two things previously believed to be equivalent in fact non-equivalent.

Imagine a train of a certain length seen from two different perspectives on the ground. The apparent length as seen from each of the two perspectives will naturally vary. If the train is in fact 10 miles long but you view it from directly behind the caboose, it will appear to be only 11 feet long. If you view it from the side, it will appear much longer. Have you discovered an instance of A=A being violated? No. What you have discovered is that the apparent length from perspective A is not equal to the apparent length from perspective B. The situation is more clear as seen from high above. Einstein's theories give us that different vantage point where we can see just how the two perspectives are related and how they don't in fact disagree about any objective actuality. The mistake was in thinking that the apparent length from a given ground-based perspective was the actual length and that therefore, all apparent lengths of a given train must be equal.

If the law of identity is just a matter of convention and doesn't say anything about physical reality, then we are quite lost! The fact that logic and math are so applicable to physical modelling suggests pretty strongly that they aren't merely conventional. They have something to do with physical reality, surely!

It seems to me that mathematics is just an abstract way of stating universal truths that physical reality must obey. Physical reality cannot, for example, involve an actualized contradiction. Perhaps you could say that math is all about the laws of substance itself.

Didn't we get our math in the first place by abstracting from or generalizing about physical reality? We started making marks to stand for physical things. Obviously, the particular marks and sounds we make are simply a matter of convention. One could imagine, for example, that an alien civilization has a different way of representing these universal truths. But would you suppose that aliens could have a mathematics or logic that is simply irreconcilable with ours?

Didn't we get our math in the first place by abstracting from or generalizing about physical reality? We started making marks to stand for physical things. Obviously, the particular marks and sounds we make are simply a matter of convention. One could imagine, for example, that an alien civilization has a different way of representing these universal truths. But would you suppose that aliens could have a mathematics or logic that is simply irreconcilable with ours? — petrichor

With relativity, it requires a conscious observer with a particular location. Aliens are conscious as well. Without observers, would “location” even make any sense?

It's just that that's the language we're using to describe phenomena.

If we used a "natural language" instead, we could say, "All known laws of science are in English."

It's just like the world being in "black & white" with black and white film, or the world being only shades of red/pink/white if we only have red and white paint. — Terrapin Station

Not really. Mathematical structures are pure abstractions I believe unlike natural language. Yet they seem to apply to the real world. I'm not sure on this but you've made a good point.

With relativity, it requires a conscious observer with a particular location. Aliens are conscious as well. Without observers, would “location” even make any sense? — Noah Te Stroete

I am not sure questions about consciousness and whether relativity requires it are particularly relevant to the topic of this thread. If we pursue those questions, it would likely prove a distraction.

My point was that mathematical relations require a mind. “2+2=4” does not exist in the physical universe outside of minds.

You’re right, though. What I said may not deal with the OP. I guess ”2+2=4” is always true (universally true). It’s just that it’s not possible without minds.

The way we talk and think about the truth that something is identical with itself or simply is itself is a convention that perhaps requires our minds. Maybe all abstractions require minds. But something's actually being identical with itself, or the impossibility of it being not itself does not seem to require our minds. But the abstractions we make seem to reflect something about the world in itself even though abstractions as we make them are not exactly found out in the world. The whole problem of universals is actually rather perplexing.

But let's not get derailed talking about consciousness here. That's a whole other issue. What we are talking about here is really whether or not the law of identity gets violated, whether 2+2 can ever equal something it isn't equal to.

I don't think the OP has properly demonstrated 2+2 not equaling 4. I'm in the process of trying to exactly identify the problem with the attempted demonstration.

I now realize what I originally commented with was something of a non-sequitir. You seem to know much more about maths than I do.

This means that, for velocities, 2 + 2 does not = 4, instead 2 + 2 = $\frac{4}{1+\frac{4}{c^2}}$.
This has been investigated and found to be true. — Crazy Diamond

The difference occurs because the velocities are being added from different inertial reference frames.

Suppose I am standing on a train platform and the train is travelling at 20km/h from my reference frame. If it goes 10km/h faster (from my reference frame), then it will be travelling at 20km/h + 10km/h = 30km/h from my reference frame. Standard addition.

However if a person on the train subsequently moves forward at 2km/h in the train's reference frame, then that person will be travelling at less than 32km/h from my reference frame.

The formula just shows how velocities from different inertial reference frames should be added.

In your original post, I believe there are some problems with your analysis. I don't know how to format the math expressions properly here like you did, and my math and physics are both very weak, so please bear with me as I fumble about here.

When you say that 2+2 = 4/(1+4/c^2), if we take your expression just as you put it, with no units, no vectors, and so on, if we do a bit of simplification here, if my rusty basic algebra skills have not led me astray, we can show that you are saying that 16/c^2 = 0, or even 16=0, which is not true. Such a result shows not that the law of non-contradiction and law of identity are wrong, but rather that there is something wrong with your equation.

I looked up velocity addition and the formula I found is different from yours. Have a look here:
https://web.pa.msu.edu/courses/2000fall/phy232/lectures/relativity/vel_add.html

You seem to be saying that u+v = (u+v)/(1+4/c^2)

And you then put in 2 for u and for v. This is apparently where you get 2+2 = (u+v)/(1+4/c^2). But this seems not correct. v', not u+v, is what equals (u+v)/(1+4/c^2).

In the relativistic theory, v' does not equal u+v, so you can't put u+v on the left side of that equation in place of v'.

It seems that you are trying to make equivalent the Newtonian and relativistic formulas, as follows:



and



and therefore, you seem to be saying, using the first formula to replace v' with u+v in the second,

u+v = (u+v)/(1+4/c^2)

But this isn't correct since the Newtonian and relativistic formulas are different as they involve different considerations. The v' in one formula is not the same as the v' in the other. So you have never shown that 2+2 equals anything other than 2+2 or 4.

And we aren't talking about simply adding two velocities. The apparent times and distances in question are different with different frames of reference.

As for physics, aren't conservation laws just the same sort of thing? This much stuff is always this much stuff. It can't be not equal to itself. No matter what you do, you can't make it not equal to itself. — petrichor

Thank you for your interesting and intelligent discussion Petrichor. The problem I have with it is that the whole point of coming up with the Einstein-Poincaré equation in the first place was to explain the strange but true observed fact that c + c = c. By all of your discussion it would be absurd for that to be true, but it is, and it is precisely that absurdity that led them to come up with the solution. When you add c and c together using that equation and not c + c = 2c, you get the required answer, that c + c = c, which is true. And that's the observed fact that underlies my whole discussion.
I've been thinking about this all day and I think the answer might be that it is not the reference frames that matter, but that no actual 'thing' is being added when you add velocities. Taking two apples and physically putting them with two apples and counting the resultant number of apples is qualitatively different from a moving object accelerating and doubling its previous speed, or something travelling at the speed of light emitting light which is observed by someone at rest relative to the light as travelling at the speed of light. In these instances, nothing is actually being added to anything in reality, so that saying that going from 2 mph to 4 mph is 'adding' 2 mph to the original 2 is just a human interpretive parsing of a phenomenon that didn't actually involve any addition. So that far from finding a genuine counter-example, I have confused two things that are different, made a category mistake perhaps by saying that adding velocities is the same type of thing physically as adding apples, when it isn't. I don't know, I'm not sure about this at all, I'll have to think about this one a bit longer...

Suppose I am standing on a train platform and the train is travelling at 20km/h from my reference frame. If it goes 10km/h faster (from my reference frame), then it will be travelling at 20km/h + 10km/h = 30km/h from my reference frame. Standard addition. — Andrew M

If you are standing on a train platform and the train is travelling at 20km/h according to your reference frame, and the driver switches the headlamp on and emits light at the speed c, then in your reference frame you will measure the light as travelling at c, and not at c + 20km/hr. That was the puzzling observed phenomenon in the Michelson-Morley experiment that led to Special Relativity in the first place. If something is travelling at the speed of light and it emits light, everyone will measure that light as travelling at exactly the speed of light, whatever their reference frame. No standard addition ever works with light. c + c = c, and that's the observation that led to the equation. I don't see how and why it can be dismissed as only a superficial feature of different reference frames. Please enlighten me! (Pun intended...)

My physics and math education are so limited and rusty! I am not sure how to deal with this. I've been trying to read and think about it, but I am just way out of my depth here. And my mental blocks are difficult to surmount.

Is "c + c = c" actually a valid expression? Do physicists actually ever put it that way? If they do, we should be able to find a discussion about this oddity somewhere. I don't even know how to Google it. I tried and didn't come up with anything relevant. If you can point me somewhere, I'd appreciate it!

I noticed that if you plug c in for u and for v in that equation, the result is just c. But we aren't really putting c+c on the left side of the equation, are we? It seems obviously problematic to do so, as u+v simply does not equal (u+v)/(1+4/c^2).


There simply must be something wrong with looking at it this way, but at the moment, I just don't have the requisite understanding to see what it is.

You could read Einstein’s paper on Special Relativity. It’s short.

Isn't the whole reason for the new equation that you can't simply add relativistic velocities as in the old equation? In relativity, c is a relativistic velocity. And you can't add relativistic velocities directly as in c+c. To put c+c on the left side of the new equation would be to try to make the old equation equivalent to the new one, which it isn't. Right?

Play with this graph I made here:
https://www.desmos.com/calculator/yejpszq9ev

y = (u+v)/(1+4/x^2)

Now test to see if y ever equals u + v.

I put in x instead of c to show that no matter the value of x, for any sum of nonzero values for u and v, y never equals u + v. As the value of x increases, the y value asymptotically approaches u+v, but never reaches it.

I suppose I'll need to! But does it put things that way? I'll have a look.