Did abstract mathematics, such as Pythagoras鈥檚 theorem, exist before the big bang?
Simon McLeish
Lechlade, Gloucestershire, UK
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The notion of the existence of mathematical ideas is a complex one.
One way to look at it is that mathematics is about the use of logical thought to derive information, often information about other mathematical ideas. The use of objective logic should mean that mathematical ideas are eternal: they have always been, and always will be.
Pythagoras鈥檚 theorem holds true now, was true before the big bang and will be true when there are no human beings to understand it. Secondly, it could be argued that without an intelligent mind able to comprehend it (and prove it), the theorem can鈥檛 be said to exist. In other words, until someone came up with the concepts involved (lines, angles and the relationship between them) and put them together, the theorem effectively didn鈥檛 exist. The actual proof of Pythagoras鈥檚 theorem probably originated before Pythagoras, and Pythagorean triples 鈥� three numbers that could be the sides of a right-angled triangle 鈥� are an older concept (but still more recent than the big bang).
A third possible opinion is that pure mathematics doesn鈥檛 really exist at all 鈥� never has and never will. While mathematicians tend to use visualisations to view mathematical 鈥渙bjects鈥� such as sets and functions, mathematics is about abstracted objects that ignore reality. For instance, a line without width couldn鈥檛 exist in the real world, but that is a property of a line in geometry, and the numbers that are used to measure the sides of a right-angled triangle have a mathematical precision far finer than could be obtained from a real-life triangle.
Perfection is the soul of mathematical thinking, but it never exists in the real world.
Nick Canning
Coleraine, County Londonderry, UK
The big bang is a singularity in space-time, so the notion of 鈥渂efore the big bang鈥� is meaningless. Are you asking 鈥淚s mathematics a pre-existing reality or a free creation of the mathematician?鈥� Both positions have been asserted.
Plato thought mathematical truth exists in some immaterial, nontemporal, non-physical mental realm of universal forms and that the mathematician doesn鈥檛 so much discover it as remember it from when their own pre-existing immortal soul (mind) was in contact with these forms.
Rejecting this view of mind, I think mathematics is the logical working-out of necessary conclusions from sets of mutually consistent axioms, freely chosen by the mathematician. The fact that it is useful relies on the axioms you choose being a good representation of relations found in nature, which you are interested in describing.
Pythagoras鈥檚 theorem is then a necessary truth of Euclidean geometry and applies to 鈥渇lat space鈥�, but it isn鈥檛 a truth of curved space such as the geometry of the surface of a sphere. If the curvature of physical space is sufficiently small, then Pythagoras鈥檚 Theorem will be a useful approximation to the truth of measurements in that space, whatever the time of the measurement.
Roger Leitch
Bath, UK
There are those mathematicians who believe mathematics is invented, for whom the answer is no because there was nobody to write it down, and those who believe it is discovered, for whom the answer is yes.
The latter are called Platonists, after Plato and his forms. There is justification in their belief, because we can consider mathematical problems that cannot be solved, in that we cannot write a solution down, but we can prove a solution exists. Not only that, we can find properties of the solution 鈥� we may be able to answer the question, 鈥淚s the solution bounded?鈥� for example.
This approach can be useful. When looking at a problem in engineering, for example, it may be impossible to calculate all the stresses at every point in the structure of a bridge, but we can obtain an upper bound on these stresses, and the engineers can ensure everything is strong enough to withstand these stresses.
The majority of mathematicians are platonists, if for no other reason that it makes life easier.
Hillary Shaw
Newport, Shropshire, UK
Arguably yes, irrespective of what was before the big bang, if there was a 鈥渂efore鈥�. The integers are independent of anything else. We start with zero, the basis of all maths, the null set A, because zero anything is identical to zero anything else. Then we have set B, containing just the null set A, so we have a one-member set, then set C containing just sets B and A, with two members and so on. No need for a universe or a Greek philosopher or anything else. Then 32 + 42 = 52 is 鈥渁lways鈥� true, and 8,329,661 is 鈥渁lways鈥� the 560,472nd prime, and so on.
Eric Kvaalen
Les Essarts-le-Roi, France
I believe that abstract mathematics exists outside of space-time, but the question presupposes that there was such a thing as 鈥渂efore the big bang鈥�. That鈥檚 not at all certain!
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