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My question is an epistemological one. I mean "Is it possible that this extraterrestrial life have found the same models, laws of physics than ours, assuming they have or not mathematic?" Or: "Imagine Einstein meeting their best physicist, how likely is it that they came separately to the same conclusions about the universe?".

  • What do you mean by "assuming they have or not mathematic"? Do you mean that you want us to consider the role of mathematics in their ability to discover similar laws of physics? – Niel de Beaudrap Oct 22 '13 at 17:35
  • Yes, I think a first step would be to wonder how likely it is that they have the same mathematical models than ours. – Nicolas Oct 22 '13 at 17:39
  • By "laws of physics," do you mean the laws of physics that govern the universe? Or the laws of physics that intelligent beings invent to explain the universe? In the first case, all beings in the universe are subject to the same physical laws. But in the second meaning, it's a matter of history. Different civilizations could have very different sets of beliefs about the universe; and even if they were equally scientific, they might have had very different historical developments of their ideas. For example what if they are just like us but they are still in the equivalent of our 17th century? – Janet Williams Oct 22 '13 at 21:37
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I don't think there's any realistic possibility of coming to an accurate answer without having met any technologically advanced civilizations which developed independently of ours. The best we can do is think of what factors would affect the chances of having developed similar models of physics.

At issue, in part, is the question of why mathematics is effective in physics. Does mathematics have in some meaningful sense a metaphysical role in the way that the world works — is there a 'music of the spheres' whose sheet music is written in mathematical notation? Or is it that mathematics is a sufficiently versatile tool that we find it capable of describing many things, including in particular the relationships we see in the universe (in particular because these relationships prompt the development of mathematics)? The reason why this is important is that it may have ramifications for how many different approximate descriptions there are for the laws of nature using mathematics.

Consider a toy example from mathematics itself, in which we try to describe a real number 0 < x < 1 by a sequence of approximations or expressions (e.g. rational numbers which are close to x), where x is described by an infinite sequence of digits 0.x1x2x3... in whichever positional system you like. However, we suppose that you only have access to as many of the digits xi as you ask for. You can continue asking for more digits, but you can never know all of them at once.

  • If x is itself a rational number, there is a single best rational approximation, which is x = n / d itself. As soon as you obtained the "approximation" of n / d for x, you would know that it was a good approximation; and and no matter how many digits of x you asked for, it would remain a good approximation while others would become more and more noticeably wrong. In fact, using continued fractions, you could actually obtain the correct values of n and d exactly; you would never know that they were perfectly correct (you might be corrected at the very next digit that you asked for after all), but once you found them you would never have to revise your estimate for x.

    This is analogous to a material universe in which the laws are very simple: if one used the right techniques, one could eventually obtain a law of nature which never needed correcting — although you wouldn't actually know that it would never need correcting.

  • If x is a 'quadratic' irrational number, that is a solution to a polynomial ax2 + bx + c = 0 for some integers a, b, and c such that b2 − 4ac is positive but not a perfect square, then you will never arrive at a final rational approximation (because x is irrational). But if you were to use continued fractions to try to obtain a rational approximation, you would notice that the continued fraction representation eventually starts repeating. If you recognised the pattern, you could arrive at an exact representation of the irrational number — again, without ever knowing for certain that it was correct, but if you did find it you would never need to correct it.

    This is analogous to a material universe in which the laws are not extremely simple, but where with the correct techniques you could still discover a set of laws which perfectly reproduced observation up to measurement precision.

  • If x is a Liouville number, then not only is it irrational, but it is transcendental: it is not the solution to any polynomial equation over the integers. And yet it will look very, very much like a rational number, because by definition you can find rational approximations ni / di which are not only as close as you like (which you can do with any real number), but in fact you can find approximations for which there is very little room for improvement without using a much, much larger denominator D ≫ di . Thus when looking for approximations to the number, you can get very close while being subtly wrong, and without insight as to where the value comes from, little improvement is likely.

    This is an analogue to a material universe which can be modeled well by very simple laws, but where a difference in the way in which you investigate the number may give rise to different formulations of the laws, whose differences will not be very significant but where the formulations themselves may be noticeably different.

  • If x is uncomputable, e.g. if its digits are in effect completely random as though they were generated by perfectly fair coin-tosses, then different approaches to evaluating the number may give rise to significantly different approximations of x — not in the sense of having wildly different values, of course, because after all we are talking about approximations of a single number, but the expressions for these approximations may bear little in common with one another.

    This is an analogue of a material world which is rich in complexity and information, where one may approximate the physics as closely as you like, but where there is no final theory and no particular reason why refinements of the theory should look superficially much like one another.

The point of the analogies above is to give some idea of the range of possibilities (and there are likely more possibilities than those I have mentioned above) for how the physical world may be like. If you accept the analogy, then you may try to consider your question by considering giving several numerically gifted thirteen-year-olds a single digit string 0.x1x2x3... , and asking yourself: under what conditions are they likely to arrive at the same expression or approximation for the number x which it represents?

  • If x is rational with a small denominator — or a very simple Liouville number — then it is likely that many of them will arrive at the same rational expression or approximation in a short amount of time.

  • If x is rational with a very large denominator, a somewhat complicated Liouville number, or a quadratic irrational number, then individual insights or the techniques which they use are likely to become significant in what approximations they produce.

  • If x is a more complicated algebraic number, or a result of coin-tosses, then the particular techniques they use is likely to be very important to determine what approximations they obtain, and there is no particular reason why they should have the same estimate.

What I have tried to suggest above is that even within mathematics itself, there are multiple approaches that one may take to investigating approximations to some number. It is true that some methods are easier than others, and for that reason may be easily reproduced by independent ingenious minds. Whether or not these easily-reproduced ideas are enough to find the pattern of the number to be approximated is the crux. There exist numbers which simply have no pattern, and also numbers whose pattern is more complicated or harder to extract than others.

The same is true of conceivable physical laws. As an apparatus for describing relationships, mathematics can describe any pattern worthy of the name, but there is no reason why everyone should be so lucky as to stumble upon the same description, or even on descriptions of the same accuracy, unless the universe happens to be quite simple. And we can never know for certain that it is very simple — for after all, like a Liouville number which 'masquerades' as a rational number, any long-standing simple law may be subtly incorrect in a way that just the right nuanced insight from a different point of view might be able to anticipate.

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Mathematics is considered a universal language because the principles which it holds are the same no matter what. 10 of something is 10 of that thing no matter where you are, what you speak, and how you write it. This is the reason why the SETI project has been broadcasting mathematical formulas on multiple bases ensuring that the recipient will understand at least one of them if not all of them like we do. Because math is a fundamental building block foundation, any civilization or lifeform will have to have a form of it. This means that at that level, they will have math and can communicate back to establish a cipher for bilateral communication.

They can choose base number system 2 binary (how computers communicate) 3 ternary 4 quaternary 5 quinary 6 senary 7 septenary 8 octal 9 nonary 10 decimal (how we commonly refer to real numbers) 11 undenary 12 duodecimal 16 hexadecimal (again commonly used in programming and computer hardware) 20 vigesimal (Aztecs and Mayans used this for planetary positions) 60 sexagesimal (used by the Babylonians and still used to measure time) and even if they use one that is not here, when we send them the same value on all of these bases, they can use it to calculate what it would be on their math base and send it back allowing us to figure out the base and establish a key.

This is absolutely critical in foundation of language and basic communication with unknown languages, similar to the techniques used to decipher hieroglyphs, ancient Persian tablets and so on. This is how we decipher dead languages on our planet, so it will be applied to any new intelligence in the universe and if they are intelligent enough to be communicating with us or even receiving us, then they have reached a development stage allowing them to comprehend and adapt. Now that being said, given that math and geometry are steps toward understanding elements of physics, they would have come to the same conclusions as we have over the years. They may have come to them at different order, with different level of specificity and by different names, but the principles of the physical universe would be observed.

Now barring a silicone based lifeform able to live on the surface of a red giant and breath sulfur, we can safely assume the same rules of physical universe would apply to them as well.

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1. "Is it possible that this extraterrestrial life have found the same models, laws of physics than ours, assuming they have or not mathematic?"

It is difficult to see how extraterrestrial life would not come up with the same mathematical patterns. A sphere is a sphere is a sphere, although it can be mathematically represented in a number of different ways. There is a key, though: some descriptions are more compressible than others.

Some folks don't seem to know this (perhaps they are scientific realists and take it too far?), but we can say that the sun and planets rotate around the earth, by modeling motions with epicycles. The term is often used in a derogatory fashion, but if we think of them as simply Fourier series, there is no problem! A big problem with epicycles is that the math is harder, so we prefer a representation that is easier on the brain. Even if the brains of aliens were different from ours, it would make sense for them to adopt efficient representations of math—both in terms of cognition and [digital] computation.

2. "Imagine Einstein meeting their best physicist, how likely is it that they came separately to the same conclusions about the universe?"

It depends on whether you mean mathematical conclusions or metaphysical conclusions. We aren't guaranteed that the aliens will have had an equivalent to Democritus, who came up with a kind of atomic theory long before science could verify it. Stealing from a question on category theory:

The hierarchy of categorical doctrines: regular categories, coherent categories, Heyting categories and Boolean categories; all these correspond to well-defined logical systems, together with deductive systems and completeness theorems; they suggest that logical notions, including quantifiers, arise naturally in a specific order and are not haphazardly organized.

So it makes sense for mathematics to have similarities. We may find a similar rule for systems of metaphysics, but I don't think any is known at this point.

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Their laws of physic would only be different if they were different to us. Since the questions says that they have the same coginitive abilities as humans, their laws of physics would not be different. however, they are aliens, so they are different from us, therefore their laws of physics would be different to us. Nobody or anything is the same so the ways hich Issac Newton discovered gravity would be different to how aliens understand gravity. They may not even know what gravity is or have some other reasoning for it. So uless the aliens are exactly the same as us, like attitudes, lifestyle their body, etc. they would not have the same physic laws as we do

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