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I am particularly curious on how one can closely talk about truths and facts with the areas of knowledge mathematics and science.

I cannot seem to distinguish between these two terms with respect to mathematics especially, as I cannot even properly define or tell the difference between truth and fact when it comes to mathematics. Is mathematical truth the same as mathematical fact?

In science as well, I get the notion of scientific facts which can be produced from verification through repeated experimentation, but when does knowledge produced in the sciences become scientific truth?

For me personally, truth seems to be more powerful in form than facts but I am not entirely sure and want to try and get some perspectives here on how one can properly define truth and facts with respect to Mathematica and the sciences. And how one can distinguish the two if there exists any in both of these areas with respect to truth and facts.

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  • Mathematical truths are known as a priori, that is, necessary for experience (see philosophypages.com/dy/a5.htm#a-pr), and scientific truths are known as empirical truths, that is, true according to experience (right, that are coherent with the senses). Notice that empirical are not necessarily coherent with a priori truths.
    – RodolfoAP
    Dec 6, 2021 at 5:29
  • Thank you for your ideas, then how about facts here, how can I associate facts with those two areas? Dec 6, 2021 at 5:47
  • I'm not aware of a significant distinction between 'truth' and 'fact' being commonly made in maths or philosophy of maths; both are relative to a given formal system (e.g. ZFC+FOL), and I've not encountered anyone quibbling over whether to describe a theorem within a system as a 'truth' or as a 'fact'. That said, it is important in mathematics to distinguish between 'syntactic truth' and 'semantic truth', not least because of Tarski's undefinability theorem: en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem
    – Alexis
    Dec 6, 2021 at 6:01
  • See Facts: "Facts, philosophers like to say, are opposed to theories and to values and are to be distinguished from things, in particular from complex objects, complexes and wholes, and from relations. They are the objects of certain mental states and acts, they make truth-bearers true and correspond to truths, they are part of the furniture of the world." According to this point of view, a facts is a "piece" of the world that corresponds to a true statement. Dec 6, 2021 at 8:45
  • If we apply it to mathematics, we may say that a true arithmetical statement (i.e. a theorem) like e.g. "1 < 2" expresses a fact about numbers. Dec 6, 2021 at 8:49

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Short Answer

In common usage, fact and truth are often to express a strong degree of synonymy. In the philosophies of mathematics and science, things become a little more complicated because they are closely related, but not necessarily the same thing. For example, some thinkers might argue there are facts, but that they do not express truths, such as the scientific instrumentalists. As such, the philosophies of mathematics and the sciences host a variety of opinions on the relationship between facts and truth, depending on things like which theory of truth is considered.

Long Answer

From MW's entry for fact:

Essential Meaning of fact
1 : something that truly exists or happens : something that has actual existence Rapid electronic communication is now a fact.
2 : a true piece of information The book is filled with interesting facts and figures. Those are the (cold) hard facts of the case.

In philosophy, as always, things get more complicated. In WP, the article on Fact:

In philosophy, the concept fact is considered in epistemology and ontology. Questions of objectivity and truth are closely associated with questions of fact. A "fact" can be defined as something that is the case—that is, a state of affairs.10

Facts may be understood as information that makes a true sentence true.8 Facts may also be understood as those things to which a true sentence refers. The statement "Jupiter is the largest planet in the solar system" is about the fact Jupiter is the largest planet in the solar system.9

The Epistemology of Fact and Truth

Fact and truth are tremendously contentious notions in philosophy, period. Facts can be viewed in light of the objective/subjective dichotomy or one might prescribe to a more complicated theory such as intersubjectivity of which there are flavors, such as Daniel Dennett's heterophenomenology. Often one sees philosophers referring to true and false facts to muddy the distinction. Is the current King of France bald? It seems like a factual matter, because it is not an opinion. In a room full of scientists, there's likely to be agreement that he is or isn't. But the question presumes someone who doesn't exist. Does this mean it's not a fact, or that it's somehow nonsense? If it's nonsensical, how come it seems so meaningful? Questions like these drive debates about what truths and facts are, exactly.

Also, truth has several prominent theories such as those based on correspondence, coherence, and deflation. So, when these different epistemic theories are part of different epistemic models, one starts to get variations on what is the relationship between fact and truth. Is it a fact that a wavefunction describes a quantum state of a physical system? Post QM, Sure. But is that same claim a truth? Does a probability function correspond to an actual wave in the universe? That would seem to suggest that mathematical objects compose physical objects. That might not work for eliminative materialists who think probability waves are products of the mind and the mind is just an illusion. You get the idea.

Fact and Truth in Science

Let's take two general perspectives in science, the debate between scientific realists and instrumentalists. A realist believes that objects exist independent of observers, and that they have definitive properties. One that might be considered is locality; objects need to touch to interact. But then what does one make of non-locality which is also known as action at a distance? An instrumentalist can avoid any worries that a realist cannot by simply proclaiming that action-at-a-distance is not a property that inheres to an object so much as a concept we use as a tool to describe what we observe. It is a fact about entangled particles, but not a truth. This is a tough sell to realists. Einstein's famous quotation about God playing dice with the universe illustrates the reluctance to disentangle fact from truth.

Fact and Truth in Mathematics

A same sort of metaphysical debate occurs among nominalist, conceptualist, and realist camps in mathematics. Mathematicians since Pythagoras and Plato have wrestled with the same sorts of challenges. J.R. Lucas in his The Conceptual Roots of Mathematics shows in his first few chapters how philosophical thinking about mathematics seeks some sort of justification for mathematics. He covers in some detail how Euclid's parallel postulate can be stated equivalently or discarded in favor of others, a historical occurrence that surrounded the development of non-Euclidean geometries. This mathematical development led to changes in how physicists, who are often radical materialists, view the universe, which was transformed from one based on Euclidean geometry and Newtonian mechanics to one that embraced non-Euclidean geometry and was enriched by Einstein's relativistic mechanics.

Is a geometry and its first principles, be they truths or postulates, a property of the universe? Do those axioms constitute facts about that universe? Or are they vacuous symbols that are factual because they do not contradict each other and seem logically independent, but make no actual claim about material? Plato was more a rhetorician than a mathematician and invented the deductive argument to try to create a certain path to truth, but even he was aware of it's limits. Valid deductions can be facts, but that doesn't make them truths.

Summary

There's a lot to the question you asked, and many working mathematicians and scientists are ignorant of their own philosophical presumption. Professional mathematicians are overwhelmingly Platonic in their beliefs about facts and truth, but it is a fact that their view of math isn't the only view about what constitutes mathematical epistemology. Science often follows a similar trajectory with a wildly divergent set of views that emerged under debate from the demarcation problem of science. Therefore, a sophisticated thinker in science and mathematics makes use of the fact that there are a wide variety of positions that are available about the relationship between fact and truth, and understands the strengths and weaknesses of each position, coming to their own mind about the matter.

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Let me start by suggesting (dissatisfying as it may be) that the terms 'fact' and 'truth' are mediated and politicized to the point that they are functionally useless. When these terms are applied in conventional discourse, they almost invariably represent a cognitive (psychological) boundary beyond which the conversation is not allowed to go. Used that way the terms represent a social power dynamic, not a philosophical (ontological, epistemological) condition.

I mean, whenever I hear those terms in regular conversation my first thought is:

This is something the speaker is deeply attached to, and ought to be approached delicately so as not to provoke an emotional reaction.

That does not bode well for further discussion...

As I see it, the ideological concepts of 'fact' and 'truth' are ossified remnants of the more subtle philosophical concepts of 'observation' and 'understanding'. those break down roughly as follows:

  • An observation is an interrogatable but non-deniable event or experience, without any real reference to meaning. People often speak of observations as 'bare facts', but calling an observation a 'fact' generally removes any possibility of questioning or interrogating it. For instance, if i were to say

    a chicken crossed the road

    this would be an observation. One night question (interrogate) whether what I saw was actually a chicken (and not, say, a velociraptor), or whether the thing that was crossed was actually a road (and not, say, a dirt path or train tracks), but one would not normally question the existence of the event or experience itself. In physics an observation is a measurable data point; in math an observation is a calculation; in philosophy an observation can be a thought or a behavior or a generic event...

  • By contrast, an understanding is a claim about a class of experiences or events, one that ostensible extends to all unseen members of that class. It's often an effort to answer a 'how' or 'why' question. Thus if I say:

    chickens cross the road to get to the other side

    That would be an assertion about the motivations or intrinsic nature of chickens. As a rule, we cannot 'observe' the intrinsic nature of things, we can only reach an understanding about the intrinsic nature of things by developing models. In mathematics, these understandings are called theorems, axioms, conjectures, and the like; in physics they are usually called theories (distinct from theorems, which are more concrete, pragmatic implementations of theory); philosophy tends to use terms like principle or argument.

In general 'truth' points at something general and universal, 'fact' points at something singular and specific, and both terms are meant to give an impression of solidity and regularity. we just have to be careful to hold the terms lightly, so that we do not confuse our perceptions of the world with the world itself.

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Fact is a proposition that is (or was) known to be true (or false):

  • Shannon went to the supermarket.

Truth is more complex, as there are numerous theories, but let’s use Frege’s simplistic (Platonist) view. ‘The Truth’ and ‘The False’ are logical objects to which truth-functional sentences refer.

Putting the two together, a fact is a statement known to refer to Truth.

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