# Are truth values of all mathematical statements immutable?

Are there some mathematical statements whose truth values are not fixed, but can change? Probably something like 1+2=3 will always be true, but are there at least some mathematical statements whose truth values can change over time?

• According to the mainstream view about mathematical theories, NO. A true statement about e.g. numbers has a definite truth value (either true or false) that does bot change over time and irrespective of the fact that we know it (i.e. that we have discovered a mathematical proof of it). Dec 1 '20 at 12:26
• Note that for example in the intuitionist philosophy of Mathematics mathematical statements need not have a defined truth value. One there is a way to construct a proof for them, then they are true; prior to that, they are assumed without truth value. In this sense, the truth values are not fixed but change over time. Dec 1 '20 at 17:53
• @Conifold You should probably post your comment as an answer. It answers my question well. Dec 2 '20 at 17:06

What counts as a "mathematical statement" is somewhat ambiguous. Do we count any statements that mention mathematical objects or concepts, or those that mention only mathematical objects and concepts, no concrete/real ones. On the broad reading (that people often take in everyday discourse, as in "I hate math"), mathematical statements can easily change their truth values, "the number of planets in the Solar system is a power of 2" changed several times throughout its history.

But this is not the typical approach in philosophy. In the closely related analytic/synthetic distinction what counts as analytic is that which depends only on conventions ("meanings"), anything that involves factual information is synthetic, even if it also depends on conventions. Although treating mathematics as purely conventional is controversial and, strictly speaking, inaccurate, it is still about abstractions, like sets and numbers, whether we think of them as conventional or not. Abstract entities are understood by most as causally inert and timeless in some sense, and that means that truth values of statements about them can not change, see Truth-value realism.

Of course, our conventions, i.e. definitions and/or axioms, can change. The concepts of "function" or "continuous function" do not mean today what they meant in the 18th century, see Function Concept on MacTutor. Given that, the truth value of the sentence "continuous function is differentiable at most points" did change from true to false. But most people would interpret that as equivocation, the same words simply refer to different things, so we are dealing with different statements. Even if statements stay roughly the same our beliefs about their truth value can change as well. Until the 19th century almost everybody believed that "the parallel postulate is derivable from the rest in Euclidean geometry" is true. But, on the majority view, its truth value was always false, we simply found that out only after hyperbolic geometry was discovered.

However, there are schools of thought, especially associated with confirmation holism and cultural relativism (Feyerabend, Kuhn, Bloor, French feminists), that see things differently, see e.g Bloor's Knowledge and Social Imagery. They see mathematical "truths", like all other "truths", as subsisting on transient paradigms or cultural conventions, and "timeless truths" generally as an exercise in mythical platonism or worse, the "tyranny of truth" with ulterior motives. On their view, abstractions come into existence with cultures that practice their use, and wink out of it with their passing. When conventions governing the use of abstractions change there is no platonic realm to pin their old "meanings" to, and so we would have to say that statements about them change their truth values as well, if they even still make sense.

• Of course, our conventions, i.e. definitions and/or axioms, can change. The concepts of "function" or "continuous function" do not mean today what they meant in the 18th century -- I beg to differ. At least if you mean what I think you mean, because describing it it terms of "meanings" and "conventions" muddles the point. It much more useful to think about concepts and words, which denote them. Then it's it easy to see that while it may take time to refine our description of concepts and agree on the what concept is denoted by a particular word (e.g. continuous vs differentiable).... Dec 5 '20 at 3:50
• ... the concept themselves are timeless -- they do not change. And they didn't -- for example, Euler's understanding of continuous function was no different from ours, according to the MacTutor page that you've linked. What might have been wrong is his theorem claiming that if a function is defined in a certain way, it must be continuous. Same as with Euclidean geometry. It is true now as it always has been -- as a special case of hyperbolic geometry. Dec 5 '20 at 3:56
• @YuriAlexandrovich Euler's concept of function was "an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities". Modern conception, traceable to Fourier and Dirichlet, is quite distinct:"a succession of values or ordinates each of which is arbitrary". Given the 18th century idea of "analytic expressions" much followed from the first that does not from the second. And neither Euler nor even Cauchy had modern "understanding" of continuity, it is due to Weierstrass, hence the controversy over nowhere differentiable continuous functions. Dec 5 '20 at 9:15
• OK, imagine an alternative history. Instead of stretching the meaning of "function" to every godawful edge case (so sought strictly for their horrendous depravity, caring little for being useless otherwise... and caring even less so for Poincaré's feelings... sending an occasional frog for Christmas hardly counts).... so imagine, instead, they would leave "function" to denote the same concept, and come up with new names for the above population? Like "map" for non-analyticals, etc? Dec 5 '20 at 17:26
• To sum it up: the "abstract" objects being timeless means they always existed, and always will. Euler's "function" existed before he described it. And it always coexists (-ed) with its modern equivalent... The way we reference them is a different story. Dec 5 '20 at 17:41

Propositions must not change within a set of rules, as per Aristotle's first law [1], which implies that a proposition must remain constant and identical on the set.

Elements that change along time (or another referential) are called variables, and its analysis of truthfulness implies including the referential. By doing that, the variable is not anymore the value of truth, but a function relating the variable and the referential.

Let me explain that a bit. Imagine that you have a variable position (x) that changes along time (t): 2 the first minute, 4 the second minute, etc. Then, x=f(t) (position is a function of time).

So, the proposition x=2 is true just for an instant, just like t=3. So, that's not the Aristotelian truth. The proposition which value of truth remains constant along all the analysis is x=2t. If you think it a bit, that a quite clever way to keep Aristotelian rules accepting variables that change along time.

are there at least some mathematical statements whose truth values can change over time?

No. Something can change over time ONLY if it is affected by time. And mathematical statements aren't, because they live in the abstract timeless universe that is Math. In it, there's neither time nor change; no movement, no life. Perfect ice crystals arranged into ever more complex concepts -- endless φ-dimensional labyrinth frozen in time.

Now, as far as Math concerned, for all its intents and purposes, it is better to assume that the crystal-palace above is real, making mathematical truth objective and, thus, universal (true for all, or for no one) -- just the way the real truth is objective and universal.

Back in the real world, we too better be real about having a telepathic link to cloud9... But, just as true, we are not doing Math, so HA! <== a perfectly valid answer for everyone, BTW.

And if only everyone could take a hint! We would live in a different, a better, world, where I wouldn't have to keep typing, and... you are still reading! why you are still reading?...

No, never mind. It's all in your head. All of it, as everyone else has a copy of everything. You have 9 and I have mine... Welcome to intersubjectivity.

## Arriving at Truth by systematic preconditioning

The evaluation of any Mathematical object to be True, can only be by preconditioning a number of other equations to be True as well. Thus in this layer of abstractions, the lowest level should be the hardest to objectively prove (i.e. f(x) = 2*x = x+1 only if x=1; thus f(x=1) = 1+1=2; that is real number twice outputs the composition of these two numbers that is 2).

## Hypothesis

The values (Truths) of hypothesis can change over time with respect to testing with different Premises. One of them is the Riemann hypothesis .