“Evidence” is often misunderstood by those who study very little of science, or those who aren’t good at science.
What is the point of trying to "educate" those whose familiarity with and understanding of scientific methods and the nature of science by presenting a fundamentally flawed, mischaracterization of the actual practice of science, the way that scientists understand and use evidence, and finally top this all off with some (at best) misleading statements about formal proofs?
I absolutely agree that often individuals in debates of a political or religious nature (for example) will engage in rhetoric involving the misuse use or inappropriate use of terms like "prove" of "proof" and so forth. Proof is (generally speaking) for mathematics. In some sciences, it is also fundamental, but the difference is that proofs in the sciences cannot take the axioms that proofs require as true by definition as in mathematics and (symbolic or formal) logic.
An appropriate response would be to point out that e.g., one cannot "prove" that there exists an external reality or really anything at all about reality as one can mathematically. An inappropriate response would be to misrepresent scientific inquiry, mischaracterize scientific methods, and restate the kinds of problematic claims that we have been trying to eradicate from science education and popular (mis)conception for decades.
Proofs are only logical models or logical statements, often expressed mathematically as equations or formulas, with combination of variables, constants and numbers.
This is not really at all accurate. In particular, the "logical models" part is hard to understand in the context of formal/mathematical proofs.
Proof theory is itself an area of mathematics, logic, and philosophy. But in the general context of proofs one finds in the mathematics literature as well as in scientific literature (among other places), it is enough to define proofs by what they are: arguments in which the conclusions must be true granted that the premises are true. In symbolic logic, the symbols in a "proof" (or derivation) can be wholly abstract and one has in general certain axioms and/or rules that allow one to move from one line in a proof to the next. In mathematics, things are not always so strict, in that proofs are often given in a combination of sentences supplemented by mathematical symbols or formal statements when appropriate and the methods by which the conclusion follows are not typically constrained to those of an axiomatic formal language or system. Put more simply, one finds quite frequently mathematical proofs that involve the derivation of a contradiction given the negation of a statement one wishes to prove true.
But the main point is that, outside of highly specialized fields in mathematics and logic, proofs do not generally need to rely on the same degree of structure or appeals to a specified set of axioms, schemes, etc.
Rather, it should be the case that it would be possible to rewrite the proof in such a system if required, starting with e.g., ZF (with or without the axiom of choice) and spending the required pages to get to e.g., the second sentence/statement in the proof.
Thus, there are several absolutely central proofs used in quantum foundations that all go by something like "Bell's theorem" (the theorem is, strictly speaking, something that is shown to be true by the proofs). These are used by experimentalists in groundbreaking research in past years and currently. They are also used in the literature by theorists. In fact, a huge number of papers in a wide variety of journals by numerous different researchers have been published just in the past few years that both use as evidence proofs of Bell's theorem and involve proofs about the nature of reality given certain assumptions (e.g., locality or no hidden variables). One example of a slew of papers, conferences, etc., would be the renewed interest in Wigner's friend by the extended Wigner's friend particularly since ~2015.
More generally, one finds proofs all over the place in certain sciences in ways that are taken as evidence that entire classes of physical theories (for example) are true or that certain methods can be used and the results be used validly as evidence.
For example, it wasn't until the proof that certain quantum field theories of the standard model were renormalizable that it became clear quantum field theory could provide the backbone of modern particle physics (and the standard models of particle physics and cosmology, respectively). Difficulties with the not yet rigorous demonstrations of the formal validity of renormalization itself remains an outstanding problem in a slew of fields.
And that's just physics (and actually a rather small number of fields in physics).
While mathematical equations (proofs) are useful tools used in science and engineering, THEY ARE NOT EVIDENCE of anything, nor are equations superior than evidence.
They are absolutely evidence. They are in fact sometimes all the evidence we really have or can even have in principle. The relevance and nature of proofs and their centrality in the sciences ranges from the fact that 1) we must rely heavily on the assumption that certain physical properties or statistical parameters or something like these are continuous in the sense of undergrad analysis (and multivariable generalizations) and thus calculus of real numbers well-defined, but no real number can ever be measured because the universe is not large enough to represent the information required to "store" a single irrational number and we have to pretend that rational numbers (all we can ever, in principle, measure) are subsets of the reals
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2) The standard model of particle physics is given in terms of the proofs of consistencies of the representations under the product of group SU(3) X SU(2) X U(1). In fact, the entire set of particles and particle type are defined because of these proofs and are given in terms of group-theoretical properties. Nor can any experiment be taken as evidence for the standard model particles without these proofs.
Sciences rely on evidence - the more there, the better - because the evidence test the hypothesis or theory, if it is “probable” or “improbable”. Proofs or equations don’t do that.
They do, actually. Quite frequently. In fact, the entire framework of hypothesis testing (including classical frequentist, Bayesian, likelihoodist, etc.) that is the basis for interpreting experimental results as giving evidence of anything is because of e.g., proofs that certain statistical parameters have certain statistical properties. Indeed, entire fields within the sciences are based on certain equations (information theory and computer science come to mind, in addition to physics).
More generally, one doesn't ever test a theory so simply. Scientific hypotheses and theories are not organized nor do they generally resemble their popular and textbook presentations. This why, for example, all the "evidence" for special relativity existed before Einstein derived it in 1905 (using equations), but nobody else had put it together. The Michelson-Morley experiments pre-dated Einstein's work by nearly 20 years. The Lorentz contraction already existed as did basically the entire theory (certainly all the physical, experimental evidence). It wasn't until Einstein promoted to the status of something like a mathematical axiom his two postulates and showed formally that they were consistent (and then later that Minkowski supplied the geometric picture) that it was possible to realize the luminiferous aether was unnecessary and the problem was instead with the equations governing the dynamics of motion in classical mechanics under the Galilean symmetries.
Evidence are the physical thing
Physical things are not evidence independent of theory. You cannot have some set of observations be "evidence" independently of specifying what it is evidence of in terms of the theoretical framework you are using and how, logically and formally, it follows that the observations are evidence.
