Mathematics vs Time

Suppose we have a person that one day states "x+3=5". The next day he again states "x+3=5". As events, we can say they are different but does the meaning of the expression has changed? It seems obvious that this isn't true but how can we prove it? Is it because mathematics are independent of time? Could we say the same for physical objects? For example, two objects are produced different days from the same factory. Would we call them the same objects?

• The point is right: physical objects are located in time while abstract ones (if any) are time-independent. Apr 4 '20 at 16:10
• @MauroALLEGRANZA Thanks for the answer. This was my intuition but I wanted to check it. Apr 4 '20 at 16:47
• You can have a mathematical system where the system has different states at different internal time-coordinates though. One could say that statements about the state of the system at each time coordinate are still "timelessly" true in some metaphysical sense, but then in metaphysics there are some views on time that say that statements about events in our future are already true or false, so in that sense are similarly "timelessly" true (see fatalism and the "B theory of time"). Apr 5 '20 at 3:58
• You don't seem to be asking about time per se, but The Problem Of Induction. Your example is sketchy too, because x is used as a variable which can be changed, so the same person who saud these things might have put x to any number of other uses in between, with no consequences for the question. Assuming x is fixed & has a single value is counter to the power of algebra. You equate products and abstractions, which is a non-starter. It's not a coherent question. Apr 5 '20 at 20:18
• There are some words the denotation of which change over time ; for example " tomorrow", or " now". But there is no such term in the sentence " x+3=5". This sentence is always equivalent to "x = 2 ", whatever time be it uttered.
– user37859
Apr 9 '20 at 16:48

Mathematical facts are timeless. They are discovered by axioms that happen to be chosen from the intuition of the mathematician. A physical object, produced from the same factory, identical in all physical characteristics is necessarily distinguished by the fact that

• They are made of fermions that cannot occupy the same quantum state
• Things are more of events in spacetime rather than solid immutable objects like mathetmaical theorems

In fact, Einstein was troubled by time in the regard that why does the "now" exist. If the natural laws are mathematical, then the results of computation are already writ in the fabric of universal logic. Therefore there would be no need for a temporal evolution. Lee Smolin uses the same idea to show that the Universe cannot have immutable laws, because if it did, the moment "now" shouldn't exist. The evolution of physical laws themselves describe a flow of time according to him.

Now here is the funny thing, nobody can prove any of those statements: if physical laws really are timeless, then there seems to be no reason why successions of moments exist, whether consciousness has something to do with this apparent presence of now and absence of the past and future remains to be seen.

• "axioms that happen to be chosen from the intuition of the mathematician" - the intuition of a mathematician doesn't sound like a very robust or scientific basis on which to assert "facts"! Apr 5 '20 at 6:05
• @Steve No it doesn't. But it happens to be so often the case that intuition is derived from physics of the universe. Take topology for example. How would a being come to know about changing shapes and preserving boundaries from crossing without first having seen them in real life? Apr 5 '20 at 6:09
• I wouldn't say intuition is derived "from the physics of the universe", but from a tacit understanding (or tacit partial understanding), sometimes based in practical experience, sometimes based in a system of concepts absorbed in one way or another from culture. In some cases "intuition" can be a cover for pure fantasy, or for beliefs chosen to serve another agenda. As for topology, the basic principle that some relational essence is preserved under stretching, bending, and so on, is strongly suggested (I would speculate) by exposure to netting, woven fabrics, and suchlike. Apr 5 '20 at 7:24

The most voted answer asserts that "mathematical facts are timeless". However, it is my opinion that in order to assert this you need to have a Platonic view of mathematics: indeed, if they are timeless, then they must exist independently of your knowledge, and in particular what is true and false in mathematics is pre-determined regardless of human mathematical activity.

A lot of the work in the philosophy of mathematics takes an anti-platonic view; already the Hilbertian school stressed more on the activity of the mathematician. But the most important work to cite, one that puts time at the very heart of the nature of mathematics, is that of Brouwer.

In Brouwerian intuitionism mathematics is essentially a mental activity carried out by the mathematician, prior to any form of formalization. In this sense, what you ask is explained in terms of the person having carried out two (not necessarily equal) constructions, that ultimately led to assert equal facts. The meaning of the written expression is understood as somebody having said "I carried out this construction"; however the written expression is mathematically moot until you carry out (or at least believe you can carry out) a mental construction establishing the indicated fact.

• I don't think this is strictly true; a Logicist view of mathematics could also argue for the independence of mathematical facts from any specific concrete objects. What you need is a Realist view of mathematics - Platonism is just the most common and easiest form of realism we tend to come across in practicing researchers of maths. Sep 26 '20 at 18:34
• @SofieSelnes although I am not an expert in logicism, I believe that a pristinely anti-realist logicism is a very difficult position to defend Sep 27 '20 at 13:53
• While I will give you that mental activity is precisely what mathematicians do, it fails to capture how mathematical "facts" are determined; For example, if the entire world starts to delude an axiom: 2 + 2 = 5, and construct logical systems based off that assertion, there might be a completely chaotic world, but such chaos could be disregarded as "randomness". Could our own delusion be the source of chaotic activity inevitable in models of nature? After all field medals are handed out only when the reviewers "trust" the axioms and the mental construction processes, whatever they might be. Oct 23 '20 at 14:25

Mathematics does not depend on time, except if it wants to :)

If we take your expression (x+3=5) , from mathematical standpoint first we must define operands (numbers 3 and 5 ) then operators ( + and =, respectively addition and relation of equality) . We must define logic to be used for evaluating truthfulness of the expression (Boolean logic/algebra is usually used, but there are others), and only after that we could talk about x. Mathematicians tend to define this rigorously, but let's assume common interpretation - in this case x must be 2 in order for this expression to be true .

As you can see, mathematics tend to invent its own rules. Mathematical axioms and subsequent theorems do not depend on anything . For example, Peano axioms simply state "0 is a natural number" etc ... without mentioning natural world, human observations, physics or anything similar. Mathematics could introduce time as an variable, so our x could be x(t), and we could have x(1)=2 and x(2)=6 . In this case or expression x+3=5 would be true for t=1 and false for t=2 . But again, from mathematical standpoint this is voluntary. Mathematics does not force us to have x dependent on t .

What about physics ? Physics on the other hand is completely natural science, in fact the name physics could be interpreted as knowledge of nature. As such, physics is obliged to take natural phenomena like time into account. Physics uses mathematics for its description of reality, but with a caveat: all equations in physics are just models of the reality or simplifications of the system, "good enough" to be used in scientific or practical purposes. In fact, this is why physics has theoretical and experimental side. While theoretical physics must provide mathematical models to be tested in experiments, it also must try to explain unforeseen or unusual phenomena encountered during experiments.

So what about our factory objects ? From physical standpoint, there are no two identical objects in universe, but for the purpose of mathematical model physics could assume them to be identical. Again, this depends on circumstances and "good enough" criteria. For example, if you are not interested in minute details, any car of the same type could be used for crash tests or handling tests. But if you want to study effects of the sun on paint, then you must select specific car, expose it to certain light amplitudes and frequencies at certain temperature etc .