I'm sure that the former's been covered in the literature, what about the latter?

Does anyone know the value of pi?

Can computers?

Allow me to explain...

Contrary to popular belief and what most mathematicians will tell you, all of the digits in the decimal expansion of π are known! They are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is the order that they appear that is not known.

If we assume that no-one and no computer can tell us all pi's digits, and further that these cannot be computed or known... can computers nevertheless use the real value of pi in calculations, and if not how can mathematicians use the value in rigorous proof ?

closed as unclear what you're asking by WillO, user2953, jeroenk, iphigenie, virmaior Jun 1 '15 at 5:24

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  • 1
    That depends. It can't be explicitly represented by finite means, so in this sense the answer would be "no." – Atamiri May 31 '15 at 14:47
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    Of course pi can be explicitly represented by finite means. It takes 10 characters to type 4 arctan (1). – WillO May 31 '15 at 15:11
  • Contrast this question with the following questions: can anyone know the value of 5, or one billion seven million and four, or negative three, or one-third, or the square root of 2?? If you can demonstrate what precisely you might mean for these questions, you may be able to make your question answerable. Just what properties (e.g. perhaps specific relationships to small integers?) make a number have a 'knowable value'? – Niel de Beaudrap Jun 1 '15 at 22:50
  • you have a very strange definition of all of the digits. You seem to confuse our counting system with something else. Just because we use the digits 0-9 to count doesnt mean this number cant be stored in 0 and 1s or in some other kind of counting system. ofc all the values of a counting system are known but that doesnt change the fact that we dont know all of the digits of pi – yamm Jun 2 '15 at 13:52

Computers can absolutely use the "exact" value of pi in the same way that humans do: by representing it symbolically and performing symbolic calculations, and only at the last step, if necessary, providing an approximate decimal expansion.

For example, Wolfram Alpha (Mathematica) knows that e^(-i*\pi) = -1. (And a lot of other stuff. Showoff.)

There is a question of whether a computer can "know" something in the sense that is typically used in philosophy for people, but this kind of representational ability isn't something that distinguishes computers and people.

  • yeah having stepped back from the question i think that this is the answer - we don't need to know every digit to symbolically manipulate the term, either as humans or in calculating machines – user6917 Jun 2 '15 at 6:10

Knowing pi as an expansion of digits is in a quite stronger sense not knowing pi at all; it's original geometric definition rests on the ratio of the circumference of a circle to its diameter; why is this even a useful definition - first because a circle is a simple geometric figure; look around you to see how many shapes are approximate circles or spheres; a square piece of paper when you increase its number of sides becomes a circle; it's also, more significantly given its provenance the shape that the sun moves around the sun.

But why should we note this ratio? Because knowing the circumference of a shape is significant; as it's diameter - so one might suppose its ratio might be significant; but then why not adding the two together or subtracting them?

Well, when we magnify a figure the ratio of its circumference multiplies by a certain value; as does its diameter; and it's this constancy that is recognised by seeing that the ratio is a constant.

But this is only the beginning of understanding the significance of the ratio pi.

In all this the digital expansion does not figure; it's of course important for computers since they represent everything digitally; but it is only a representation; and this representation is always approximate upto a precision determined by the designer.

There are systems of exact arithmetic that symbolically represent transcendental numbers (those numbers whose digits don't ever repeat); simply by representing this number by some additional symbol, as is normally done in mathematics by the Greek letter pi.

Computers only 'know' something in a manner of speaking; that is the Turing Test; which is an operational definition; which is good enough for practical and pragmatic purposes; but not good enough for philosophic purposes.

Knowing presupposes a knower; a thing cannot know; having no self to speak of; a computer no matter how closely it imitates knowing in a different sense does not know at all; consider for example a film starting Selma Hayek; it is recognisably her but one doesn't ordinarily confuse the image on the film with the person herself.

Finally, knowing pi as an unending string of digits is to only know that there is such a number; and that it is significant in some way; in a way it's a modern form of numerology; which is to mathematics what astrology is to astronomy; and in its own way, and a sociological way not uninsignificant either.