(Note: I have a degree in mathematics, but this question goes a bit beyond that.)

Take a value like pi ($\pi$). Only a mere handful of decimals is all we could empirically verify to be "true" (measuring the roundest object we could construct, for example).

What if one decimal, say the billion billion billionth one (or somewhere beyond where the current record is), actually changes with time? We could compute it with some numerical method on a computer and realize this decimal seems to differ "every" time we do the computations. Maybe a god is fiddling with it.

Could there be any practical effects, and does it make any sense? What could such a discovery possibly imply?

  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – James Jan 10 '18 at 21:21
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    $\begingroup$ Pi is is a number defined by humans based on the properties of ideal space and geometry, so the number itself changing doesn't really make sense. A better question to ask would be if the physical constants change, for example the speed of light, the gravitational constant, or Planck's constant. $\endgroup$ – IndigoFenix Jan 11 '18 at 6:22

The question you ask is not really the question you seek. The answer to what you asked is boringly simple: "No, a mathematical constant cannot change because mathematical constants are defined to be not changing." While there are some philosophical questions about whether mathematics could indeed be the underpinnings of reality, mathematics is more commonly treated as a thing we created to make sense of the universe. If we decided that π is a constant, it indeed is a constant because we defined it thus.

The more interesting question is the empirical one you are looking at. What if the mathematical relationships that are valid at one moment are sightly invalid in the next. This is a slightly more nuanced question. It's asking "If I hypothetically had a perfectly circular object, and measured the ratio of its circumference to its diameter, would it change over time?"

The limit to this, of course, is our ability to make circular objects. We live in 3-space, not flatland, so maybe a sphere is a better choice:

Silicon Sphere

This beauty is the most spherical object we have ever created. No, not Achim Leistner's balding head -- the silicon sphere in front of him. It's part of Project Avagadro, an effort to fix the mass of the kilogram to something other than a particular lump of platinum-iridium sitting in a vault in France. Liestner is the head optician of the effort, and his spheres are incredibly spherical (a 93.6mm sphere that's 35nm out of spherical -- roughly 370 parts per billion out of spherical!)

So we can see that we're not going to notice a change in a billionths place with anything we can make. We can get about 7 or 8 decimal places at the most. But what if we look bigger?

It turns out that space is big. Really big. So mindbogglingly big that even mathematicians have trouble comprehending how enormously big it is. It's also really old. Really really old. In my opinion, it's not quite as mindbogglingly old as it is mindbogglingly big, but you get the idea. Small changes have... substantial implications.

Consider two objects that are at rest in the same reference frame, with no forces acting on them (you counteracted gravity, somehow). You measure their distance to be exactly 1000mm apart. Don't ask me how you did it. Now go measure their distance apart again. Is it 1000mm? No. It's not. It's actually slightly larger. Why? Because of the expansion of space. If you were to assume that the "space" that we measure in is reality, you would find that that implies that space is growing at a constant rate. If you made your measurements 1 second apart, you would find the second measurement to be 1000.0000000000000022685455mm, give or take a few quintilionths.

What does this imply? Lots of things. For example, energy is not precisely conserved in an expanding space like this. All your energy balances are going to be slightly off. But, if you think about it, you really don't notice these effects. They are easily dominated by other effects. A classic question is "is the expansion of space causing LA and New York to drift apart," to which the answer is no. If you fixed two points in space (one over LA and one over NY at an epoch), those points in space would drift. But these effects are much weaker than the electrostatic forces holding our planet together. LA and NY will stay put... or at least stay put as much as their tectonic plates permit.

So you asked what would happen if some constant were to change slowly in the billionth decimal place or something. Would that make any sense? The answer turns out to be "Yes, it makes sense, and we have it in our reality today!" The expansion of space is a slow change in "fixed constants" that astronomers have to account for. However, practically speaking, it isn't all that important.

Of course, an open question would be what if such tweaked values were actually under the control of some external entity like a deity, with intent. In such cases, it's far less clear whether this would matter or not. It's possible that that billionth digit is all the control this deity needs to shape the universe as they see fit. After all, the universe is a big place. Billions and billions of stars. Rounding errors can add up!

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  • $\begingroup$ This is a great answer, but I think there are some distinct differences between the empirical measurement of distance in space and "empirical measurement" of exact value of pi. $\endgroup$ – naslundx Jan 8 '18 at 9:12
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    $\begingroup$ @naslundx How would you measure a value for pi empirically without measuring distances in space? Even Einstein's relativity started off by assuming one could have a perfect rod and a perfect clock to measure distances and times with without flaw. $\endgroup$ – Cort Ammon Jan 8 '18 at 14:16
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    $\begingroup$ Also worth noting is that pi is only the ratio of the circumference of a circle to its diameter in Euclidean spaces. Given that our space is not Euclidean (it's known to be distorted from that ideal), such measurements could be rather tricky. $\endgroup$ – Cort Ammon Jan 8 '18 at 15:42
  • $\begingroup$ Ah, I see your point now. $\endgroup$ – naslundx Jan 8 '18 at 16:36

It would make for an interesting story, but as Samwise said, it would break lots of physics and math. I'll quickly point out this post on physics.stackexchange. In summary: if they change, we start looking for why they change. We create new "laws" to superseded the limitations of previous ones.

So let's examine some effects of some constants changing (ignoring how or why they changed).

  1. Let's say pi changes. It get's bigger. Does that mean that everything circular get's heavier? Where did that mass come from? Actually, it doesn't just apply to macro physical objects - I'm sure some nerd can prove that any object is made of lots of little circles, so every objects mass may have just increased. And other things change such as orbits: Do our electrons now take longer to orbit an atom? Does this change nuclear physics? Probably. Does this change our planet's orbit somehow?
  2. All right, let's twiddle the gravitational constant. Crap, all the planets orbits just changed, and a couple stars either went supernova or collapsed into black holes. The amount of energy output from hawking radiation will likely have changed, so some black holes may have just evaporated. Probably no-one but the astronomers noticed unless the effect was enough to affect the climate on Earth (ie orbit became elliptical)
  3. How about we vary light speed? Your GPS just became less accurate, and your computer probably stopped working entirely with it's CPU suffering from timing errors. Some far-off objects in space just changed hue, and the part of the visible spectrum your eye can see may have changed. There'll be some funny effects in electrons-hitting-metal giving different amounts of energy, so your solar panels are probably no longer as efficient as they once were. Oh, and you just really puzzled some scientists at CERN....
  4. Let's change the decrease the charge on an electron (but not on a proton or neutron). HOLEY MOLEY, a whole bunch of metal parts just spontaneously disintegrated, and things physically changed size (slightly) as all the electrons now orbit closer to the atom.

I will add that if the constants do change value, but do so very fast, then you will probably not notice any effect. I can't see that we could have an experiement to see if the speed of light changes every femtosecond, and I can't see that it would cause any measurable effect either. However, this may well explain some of the probabilistic nature of quantum physics.

Also see Diracs Large number hypothesis.

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  • $\begingroup$ Well, but twiddle these constants on the billion billion billionth decimal place. Would that affect anything? $\endgroup$ – naslundx Jan 8 '18 at 9:13

Value of a mathematical (or any other "exact" Science) cannot change by its own nature.

All Exact Sciences in general and Mathematics in particular move from a defined set of postulates and definitions toward an ever-growing set of derived "truths" (theorems).

This has no relationship with the real world (beside fact initial postulates "look right").

$\pi$ value is computed in the assumption we live in a flat space. Different curvatures would lead to different values, as Riemann, Lobachevsky, Bolyai and others discovered to their dismay.

Truth is our "real" space is not flat and thus measuring ratio between circumference and radius will not yeld $\pi$.

This, however, cannot change the value of the constant because it is computed in the assumption we live in a flat ($curvature=0$) space.

Things are slightly different for other "exact" sciences (e.g.: Newton Physics) where certain "constants" (e.g.: Gravitational Constant) are not derived (computed), but are measured, instead.

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What is mathematics?

I'd say mathematics is the science of building models from axioms and basic logic, and of exploring their behaviour. Many of the most useful models represent the real world, but that's not necessary to make them worthwhile. One can learn from models which do not represent the real world.

What is a constant?

A constant is a shorthand notation for some term that comes up often during the exploration of a model. One does not say let's have a mathematics where Pi = 2, one says ah, there is a number which shows up often, call it Pi. In that sense mathematical constants are defined because they work.

So how does one change a constant?

By changing the axioms. In a way that's exactly what we're doing on worldbuilding with good answers and questions. "What do I have to change so that the wizards rule the country?" or "what do I have to do so that the circumfence of a circle is no longer Pi times diameter?" are both answered by going to the fundamental causes (the axioms) and changing them.

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Changing a single constant (PI or otherwise) will probably have a huge effect on the universe if that constant has an influence on atomic or sub-atomic particle. To stay related to PI think of all the atomic and molecular physics laws based on radius.

  • Strong nuclear force goes linear by the radius (somewhat)
  • Coulomb force goes by the radius squared
  • Intermolecular (such as Van der waals, dipole-dipole, hydrogen bond, etc..) forces go by r^2, r^3 r^4 and r^7

The difference between the linear, quadratic and higher order means that if you change the interaction potential (via PI) you are likely also to get a complete balance shift in how particles behave with each other. Material packing become differently, boiling points are changed, gravity is influenced, basically everything.

Although your proposed change is minute since the number of molecule effected is enormous (a cup of tea roughly contains 1 x 10^25 molecules) the effects will be enormous.

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I gave this in a comment, but realized it was getting much too long for a comment.

The key might be in the starting assumption that pi is not a rational number. It is a non-rational constant. Can there BE such a thing? Perhaps the crutch is that it is assumed that pi is a constant AND non-rational at the same time.

Really, are you positing that this does NOT have to be constrained to just the calculation of a constant? You are starting with a calculation based on specific numbers and specific operations that produce a specific result, and the result changes from calculation to calculation? Why wouldn't this then apply to ANY calculation? Basically, are you asking 'can any defined calculation lead to a different result?' I really don't think this is what you are asking.

Or are you suggesting that somehow the STARTING values change subtly? That, perhaps, something subtly changes about the relationship between the circumference and diameter of a circle? That it is the RATIO that subtly changes? That, for some reason, the measured circumference can change for a specific diameter?

In math, a rational number divided by a rational number gives a rational answer. If pi is not rational, then either the circumference or diameter is not rational. Since one of either or both the circumference OR the diameter is not a rational number, this is within the realm of conjecture. What is it that MAKES the measurement not rational?

In other words, rearrange the operation. Circumference equals pi times the diameter. If the diameter is rational, and pi is not rational, the diameter is not rational. If pi changes subtly, and the diameter remains the same, then the circumference (being the non-rational measurement) presumably has to change.

Since a circle has no beginning or no end, how do you measure it? There is always some perhaps non-zero-but-zero distance in the gap between the starting point of the measurement and the ending point. They can not overlap. See my answer about 10 equals 9.9999... If you START with a number like this, how would you calculate pi? Would it be different if you started with 10, vs 9.9999...? If pi is not a rational number, then you can not calculate it from rational numbers.

Perhaps Planck's Constant changes subtly in the very last point (or very first point) in the circumference between the starting point and the ending point of your measurement? If there were a Planck Constant difference in the gap, then one would just change the ending point to encompass it.

Perhaps, if you assume that the circumference really IS, by quantum mechanics, probabilistic and not deterministic, you might have your answer.

With a different-but-not-different non-rational starting value in the calculation, you could get a same-but-not-the-same calculated 'constant', could you not?

Perhaps this is an artifact of quantum mechanics and quantum indeterminability.

If Schrodinger's cat can be alive and dead at the same time, why can't non-rational constants be the-same-but-different?

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Practical effects? Maybe not. But it'd be a HUGE blow to science as it's most important cornerstone, the immutability of maths, is no longer true because now we have proof that π ≠ π, we can no longer be certain about ANYTHING anymore. There would eventually be a mad scramble to either confirm or disprove the effect in other situations.

Although at first it'd probably be written off as some kind of architectural bug in whatever computer system was being used to calculate that billionth digit, but some may come to see the discovery as proof that our entire existence is just a computer simulation being run by some higher power (the god you mentioned) and that we are seeing the limits of god's PC. At that point, the discovery would cease to be just an academic anomaly and would start have knock on effects in everyday life.

I'm sure you can imagine that if news broke that scientists had found proof that we were all figments in a computer program the news probably wouldn't be received too well. (but then again, Arthur Dent did't seem to bothered by it)

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  • $\begingroup$ It has already begun. One of the earliest effects of π deterioration would be the random disappearance of "v", "k", and "n" from certain words. The missing "n" before an apostrophe might become particularly resistant to editing. $\endgroup$ – A. I. Breveleri Jan 7 '18 at 20:02
  • $\begingroup$ It wouldn't make sense for the output of a computer calculation to change; regardless of whether π changes, the algorithm doing the calculation surely wouldn't. $\endgroup$ – Erik Jan 7 '18 at 20:05
  • $\begingroup$ @Erik: The algorithm might not change but the underlying hardware could no longer be guaranteed to be deterministic. $\endgroup$ – A. I. Breveleri Jan 7 '18 at 20:07
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    $\begingroup$ Mathematics is not science. They have different methods and different objectives. Mathematics is about absolute truth; science is about provisional approximations. Mathematics is by necessity the same in all possible worlds; science isn't. $\endgroup$ – AlexP Jan 7 '18 at 21:25

Trace through the following math.

  1. Let x = 9.999999999...

multiply both sides by 10.

  1. then 10x = 99.9999999999...

subtract 1. from 2.

  1. 9x = 90

Solve for x

  1. x = 90/9

  2. x = 10

Obviously x can not equal 10 AND 9.9999.....

Or can it?

So can pi have at least TWO solutions? That is, can pi be equal to two numbers at the same time? It would, presumably, not be in the middle, but it would be the very LAST digit.

Two numbers are different if and only if one can be subtracted from the other and result in a non-zero number. Subtracting two infinitely long numbers (two versions of pi, differing only by one digit) is, well, interesting.

What it implies is happening is, well, math is happening.

But on a more physicsy note, you might be interested in Further Evidence for Cosmological Evolution of the Fine Structure Constant

Interestingly, independent results are now emerging which support the trend in ∆α/α we find. The most recent CMB data are consistent with α being smaller in the past by a few percent [30]. Also, varying speed of light models, [31], are appealing because they may explain the supernovae results for a non-zero cosmological constant and solve other cosmological problems (e.g. the horizon, flatness, monopole problems) [32]. These also require a smaller α in the past. We anticipate that further independent quasar data will provide a definitive check on our results.

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    $\begingroup$ If you subtract 9.999... from 10 you get 0.000....1, an infinitesimally small difference and yes, for all practical purposes this is the same as 0....but that doesn't mean it is "true", only that our brains don't quite comprehend infinities $\endgroup$ – Lio Elbammalf Jan 7 '18 at 23:15
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    $\begingroup$ The formal phrasing for that number game is that "10" and "9.9999..." are two different strings which represent the same real number. In order to make them actually be different numbers, we need to extend beyond the real numbers. One example system is Conway's surreal numbers $\endgroup$ – Cort Ammon Jan 7 '18 at 23:31
  • $\begingroup$ @Lio Elbammalf Actually, you don't. The limit is zero. There is no number between 10 and 9.9999.... $\endgroup$ – Justin Thyme Jan 8 '18 at 5:26
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    $\begingroup$ This is a very different thing. The difference between 10 and 9.999... tends to zero (so they are merely two different representations, as stated above), whereas a change of pi at a specified (finite) decimal place yields a very finite difference. $\endgroup$ – naslundx Jan 8 '18 at 9:45
  • $\begingroup$ Please note that, in this case, it was the FIRST digit that changed, depending on HOW it was computed. But really, does what you posit HAVE to be constrained to just the calculation of a constant? You are starting with a calculation based on specific numbers and specific operations that produce a specific result, and the result changes from calculation to calculation. Why wouldn't this then apply to ANY calculation? Basically, are you not are asking 'can any defined calculation lead to a different result?' $\endgroup$ – Justin Thyme Jan 8 '18 at 14:37

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