Always I wonder about that if some day alien wants to talk to us, then how can we talk to them? They didn't know any of the language on the Earth.
What will be the basic building blocks for it?
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$\begingroup$ There might be a good question here, but as phrased it is too speculative. It is unclear what you mean by "talk through maths", or why you ask about "integrals". Of course we know nothing about mathematical systems used by undiscovered intelligences. $\endgroup$– James KCommented Feb 14, 2018 at 8:12
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$\begingroup$ It certainly worked in 1952! ;-) scifi.stackexchange.com/a/16368/51174 (in Hollywood at least) $\endgroup$– uhohCommented Feb 14, 2018 at 8:20
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1$\begingroup$ This is an interesting question nonetheless. Poincaré and Wigner might differ on how universal math might be, or if a very different civilization may use mathematics so different as it might be mutually unrecognizable vis-a-vis Earth's maths. Watch Is math discovered or invented? - Jeff Dekofsky $\endgroup$– uhohCommented Feb 14, 2018 at 8:27
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$\begingroup$ The gold plates put on board Voyagers are there to get an idea of what might work. At least if " to talk" is intended as "to communicate". $\endgroup$– AlchimistaCommented Feb 14, 2018 at 8:30
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1$\begingroup$ Weinberg's thoughts and Wolfram's (two Sevens, err... Stevens) $\endgroup$– uhohCommented Feb 14, 2018 at 8:41
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2 Answers
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Some basics:
Whether we listen for or send signals into space, something to keep in mind is that space is full of EM radiation of all wavelengths, from very long wavelength background radiation, to a range of microwave, infrared, visible and UV light from stars to gamma-rays from accretion disks and collapsing stars. Any signals we send and perhaps any signals we could receive would be like raindrops in an ocean. There's an obvious range to listen called the hydrogen line, which is the emptiest region of EM radiation in space.
Even so, across light years, signals would still be faint, so the first signal that's sent needs to be an easily recognizable signal and math is very good for that. Pi in binary or a series of repeating prime numbers wouldn't happen naturally, so signals like that are a great way of saying "We are here".
Once the first signals of prime numbers or other math are exchanged, and we see what direction the signal comes from and we know where to listen and where to send, then the communication can begin, presumably using more traditional language.
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The general plan is to start very simply and build up - would you recognise the following?
. .. ... .... .....
Numbers, right? You'd probably want to save typing out lots of dots in the long run, so you'd want to convert it to a "base-something" system, e.g.
0001 0010 0011 0100 0101
In conjunction with the original, it should be obvious that this is binary (the Aricebo message, if I remember correctly, used a form of base 10).
Then you can start describing sequences of numbers that you think the aliens should understand - primes, Pi, or as Aricebo did, the atomic numbers of H, O, C, N...the elements that primarily form life on Earth.
As long as the aliens have a concept of numbers, and a vaguely similar idea of mathematical logic to us, you can build up to most mathematical ideas in one way or another. Integrals, for example, could be done by sending a sequence corresponding to y=x^2, then defining two points in particular, and providing the resultant integration result. A civilization capable of picking up our signals should (I make no assumptions about aliens!) recognise the concept.
