Does the "unreasonable effectiveness of Mathematics" suggest we are in a simulation?

 This was a question on Quora, and I provided 2 responses: one being a comment on someone else’s post (whom I follow); and the other being my own answer.

Some years ago, I wrote a post on this topic, but this is a different perspective, or 2 different perspectives. Also, in the last year, I saw a talk given by David Chalmers on the effects of virtual reality. He pointed out that when we’re in a virtual reality using a visor, we trick our brains into treating it as if it’s real. I don’t find this surprising, though I’ve never had the experience. As a sci-fi writer, I’ve imagined future theme parks that were completely, fully immersive simulations. But I don’t believe that provides an argument that we live in a simulation, for reasons I provide in my Quora responses, given below.

 

Comment:

 

Actually, we create a ‘simulacrum’ of the ‘observable’ world in our heads, which is different to what other species might have. For example, most birds have 300 degree vision, plus they see the world in slow motion compared to us.

 

And this simulacrum is so fantastic it actually ‘feels’ like it exists outside your head. How good is that? 

 

But here’s the thing: in all these cases (including other species) that simulacrum must have a certain degree of faithfulness or accuracy with ‘reality’, because we interact with it on a daily basis, and, guess what? It can kill you.

 

But there is a solipsist version of this, which happens when we dream, but it won’t kill you, as far as we can tell, because we usually wake up.

 

Maybe I should write this as a separate answer.

 

And I did:

 

One word answer: No.

 

But having said that, there are 2 parts to this question, the first part being the famous quote from the title of Eugene Wigner’s famous essay. But I prefer this quote from the essay itself, because it succinctly captures what the essay is all about.

 

It is difficult to avoid the impression that a miracle confronts us here… or the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.

 

This should be read in conjunction with another famous quote; this time from Einstein:

 

The most incomprehensible thing about the Universe is that it’s comprehensible.

 

And it’s comprehensible because its laws can be rendered in the language of mathematics and humans have the unique ability (at least on Earth) to comprehend that language even though it appears to be neverending.

 

And this leads into the philosophical debate going as far back as Plato and Aristotle: is mathematics invented or discovered?

 

The answer to that question is dependent on how you look at mathematics. Cosmologist and Fellow of the Royal Society, John Barrow, wrote a very good book on this very topic, called Pi in the Sky. In it, he makes the pertinent point that mathematics is not so much about numbers as the relationships between numbers. He goes further and observes that once you make this leap of cognitive insight, a whole new world opens up.

 

But here’s the thing: we have invented a system of numbers, most commonly to base 10, (but other systems as well), along with specific operators and notations that provide a language to describe and mentally manipulate these relationships. But the relationships themselves are not created by us: they become manifest in our explorations. To give an extremely basic example: prime numbers. You cannot create a prime number, they simply exist, and you can’t change one into a non-prime number or vice versa. And this is very basic, because primes are called the atoms of mathematics, because all the other ‘natural’ numbers can be derived from them.

 

An interest in the stars started early among humans, and eventually some very bright people, mainly Kepler and Newton, came to realise that the movement of the planets could be described very precisely by mathematics. And then Einstein, using Riemann geometry, vectors, calculus and matrices and something called the Lorenz transformation, was able to describe the planets even more accurately and even provide very accurate models of the entire observable universe, though recently we’ve come to the limits of this and we now need new theories and possibly new mathematics.

But there is something else that Einstein’s theories don’t tell us and that is that the planetary orbits are chaotic, which means they are unpredictable and that means eventually they could actually unravel. But here’s another thing: to calculate chaotic phenomena requires a computation to infinite decimal places. Therefore I contend the Universe can’t be a computer simulation. So that’s the long version of NO.

 

 

Footnote: Both my comment and my answer were ‘upvoted’ by Eric Platt, who has a PhD in mathematics (from University of Houston) and was a former software engineer at UCAR (University Corporation for Atmospheric Research).

An impossible thought experiment

I recently watched a discussion between Roger Penrose and Jordan Peterson, which was really a question and answer session, with Peterson asking the questions and Penrose providing the answers. There was a third person involved as moderator, but I’ve forgotten his name and his interaction was minimal. It was mostly about consciousness, but also touched on quantum mechanics and Godel’s theorem.

 

I can’t remember the context, but (at point 1.06) Penrose trotted out the well-worn thought experiment of 2 people crossing a street in opposite directions, and somewhere, in some far-flung part of the cosmos, an armada of spaceships is departing for a journey to Earth. Now, according to Einstein’s theory of relativity, one of these ‘observers’ will 'say' the fleet left 100s of years in the past and the other will 'say', no, they're leaving 100s of years in the future.

I’ve always had a problem with this ‘scenario’, and I’ve discussed it previously. The thing is that neither of them can ‘observe’ anything at all, because the ‘event’ (space fleet departing) is outside the light cone of influence of Earth (in either the future or the past). So neither of them receive a signal telling them that this is what they ‘observe’. In other words, it’s something they’ve worked out with equations or a space-time diagram. Brian Greene illustrates it graphically in a YouTube video.

 

Of course, my interpretation is considered ‘naïve’ and completely wrong by Penrose and every other physicist I know of.

 

Now, some thought experiments, like the famous EPR experiment, in combination with Bell’s Theorem, can be done in the real world, and was done after Einstein’s death and effectively proved Einstein wrong (on that particular point). Another example is John Wheeler’s delayed decision thought experiment for the double-slit experiment, which was also physically done after Wheeler’s death.

 

But this thought experiment is impossible to do, even in principle. My interpretation is that you have a clear contradiction, and where you have a contradiction there is usually something wrong with one or more of your premises. My proposed resolution is that what they 'perceive' is not reality, because the event is outside the cones-of-influence (past and future) of the observers.

But let’s take the thought experiment to its logical conclusion. Let’s say the observers record their deduced ‘time of departure’ with respect to their frame of reference, and it can be looked up centuries later when the fleet actually arrives on Earth. Now, when the fleet arrives, its trajectory through spacetime is within Earth’s past light cone. The fleet has its own time record of their journey and we know how far they’ve travelled. In fact, this is no different to the return leg of the famous twin paradox thought experiment. Now observers can apply relativistic corrections to the fleet’s recorded elapsed time, and deduce a time of departure based on Earth’s frame of reference. This will give a ‘time’ which I expect will fall somewhere between the 2 times recorded by the original observers.

 

Of course, this is still an impossible thought experiment because there is no way the 2 pedestrians could know when the fleet was departing. But if a fleet of spaceships did arrive on Earth from somewhere ‘far, far away’, we could calculate exactly when it left (ref. Earth time) and there would be no contradiction.

 

 

Footnote: I know that this stems from Einstein’s discovery that simultaneity varies according to an observer’s frame of reference, and there is an excellent video that explains the maths behind it. But here’s the thing: if the observer is equi-distant from the 2 signals in the same frame of reference you’ll get ‘true’ simultaneity (watch the video). On the other hand, an observer moving with respect to the sources will not see simultaneity. A little known fact is that you have to allow for length contraction as well as time dilation to get the right answer. But here’s another thing: on a cosmic scale, 2 observers can see 2 events in opposite sequence even if they’re not moving relative to each other. BUT, if the events have a causal relationship, then all observers see the same sequence, irrespective of relative motion. (Refer Addendum 2)*

 

 

Addendum 1: I’ve given this more thought, by having imaginary dialogue with a physicist, who would tell me that my ideas are inconsistent with relativity. Naturally, I would disagree with them. I would say it’s consistent with relativity, because, for the thought experiment to actually work would require instantaneous communication, which is as contradictory with relativity theory as one can get. For the 2 hypothetical observers to ‘know’ when the far-flung event took place would require them to observe it in their ‘now’, which is impossible. So my response is strictly a philosophical one: you can’t apply relativistic theory to this situation because the 'observation' would appear to 'violate' a tenet of relativity theory. And that’s because the event is outside the observers’ light cone. Am I missing something here? 

 

*Addendum 2: I watched a video by Sabine Hossenfelder, who addresses the last sentence in the footnote of this post. She says, in fact, that 2 events that have a causal relationship in our frame of reference could appear ‘independently’ simultaneous to ‘different’ observers in another part of the Universe (watch the video). This is a variation on the thought experiment that I discuss. In practice, because there’s no possible causal relationship between the events and the far-off observers, they wouldn’t observe anything. And it doesn’t change the sequence of causal events.

 

But she’s arguing that there are a multitude of ‘nows’, in accordance with one of Einstein’s premises that all observers have the same validity. That may be correct, but why does it apply to events they can’t even observe? To be fair to Sabine, she does try to address this at the end of the video. I’ve long argued that different observers see a different ‘now’, even without relativity, but if they see a different sequence of events, at least one of them has to be wrong.

 

I want to emphasise that I don’t think Einstein’s theories of relativity are wrong, as some people do. My point is purely a philosophical one: if you have a multitude of perspectives with different versions of events, they can’t all be right. I’m simply arguing that there is an objective reality. A case in point is the twin paradox, where one twin’s clock does run slower, irrespective of what each twin ‘observes’. Mark John Fernee gives a synoptic exposition here. As he says: they each have their own ‘true time’, and one is always slower.

 

If you go far enough into the future, where the events in question fall within the observer’s past light cone, then a history can be observed. We do this with the Universe itself, right back to the CMBR, 380,000 years after the Big Bang, which is 13.8 billion years ago; both of which we claim to know with some confidence.

 

I still maintain my core point, stated explicitly in the title of this post, that, as a hypothetical, the thought experiment described is impossible to do, simply because the event can’t be observed.