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This question mostly pertains to physics and math, but I think it fits best on this site. I am not very familiar with philosophy, so I apologize if my question is not very formal.

Essentially, the idea of ambiguity/vagueness has been bothering me for many days. I see many mathematicians/physicists claiming that their theorems are completely unambiguous, but my gut tells me that nothing can be completely unambiguous (I feel that there is always a way to interpret something differently).

So, if any theorem we have ever devised is ambiguous to a certain extent (albeit a very subtle extent), how do we know what that theorem truly is? Is that theorem based on the original author's interpretation (even if he/she could not fully convey his interpretation in writing, as it is impossible to do so)? What if someone were to write a book on the theorem based on his/her own interpretation which is slightly different from the original author's interpretation? What if, in a thousand years, the original author's work is lost and we only have the secondary work? How do future generations know that their interpretation of this theory is the "right one"?

Now, I know that some people may just state that math is totally unambiguous. And, while I can't say that I've encountered any math that I've found fundamentally ambiguous, I can't help but feel as if everything humans write/say is at least a little ambiguous. These ideas have caused me a lot of anxiety recently (I am worried that my interpretations of theories are incorrect). Any perspectives on this question would be very, very much appreciated.

  • This depends on what do you mean by "thuth" and "knowledge". Usually knowledge involves some kind of predictive power. But can we know everything that will be? If my theory of cosmos is right, we can't. – rus9384 Dec 7 '18 at 7:39
  • It may be that we are all living in a simulation.. or are being dreamt by god... But we know that in only 2 centuries whatever science is.. has taken us from mud to spacedust. Science is overwhelmingly.. staggeringly.. unequivocally.. stupendously better at predicting the nature of reality than any other means of enquiry. Philosophical argument about what reality is... Fundamentally are very important... But if you have a bacterial infection.. you're going to want the antibiotics science created... Not a epistemological exposition on the flaws of empiricism. – Richard Dec 8 '18 at 0:09
  • As long as it works, who cares? Why care about the accuracy of a secondary, or tertiary, work when they add to our understanding? Or do you think history-of-philosophy is more important than any philosophical discipline? Because History-of-philosophy doesn't pretend to be absolutely accurate, only to strive for accuracy. – christo183 Dec 10 '18 at 4:09
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You are completely right about ambiguity, and that's probably a problem at many levels, mostly related to our subjectivity.

I will address, as an example, thermodynamics. Thermodynamics describes the behavior of energy in a system. It means that a system (made of parts) has energy. Take note of this: a thing, made of smaller things, has energy. But that's not evident (and in fact, that's somehow naive) for a physicist (with a critical mind): in real nature, there are no THINGS. On the real nature, there's an equivalency of things made of mass and energy. Modern definitions of quantum systems define things as portions of space.

(If you haven't read the philosophy classics, perhaps you are not aware that some philosophers - leaded by Immanuel Kant - think that space and time are just mental constructs, they are not real (they are called_ knowledge a priori of experience _). I firmly believe such point of view. I once thought that Einstein would be destroyed if he would discuss relativity with Kant. Einstein was once defeated by Bohr, and perhaps Bohr will agree a lot with Kant. But that is just a personal speculation, ala Bukowski vs. Hemingway). Then, in such case, what would a quantum system be? Just a representation of sensible facts held somewhere in the brain.

Then, perhaps only the first law (energy conserves) is correct, but only if we consider that the THING containing such energy is something that we bound subjectively (what are the real limits of a balloon of gas, since the the quarks of both THINGS can hardly be differentiated, and when elementary particles are essentially empty space?). The second law would be useless (energy spreads between the parts of the whole), since energy just changes of form, we see things, but that's just our perception. The third law (entropy is 0 at 0K) is our way of saying "hey, here's the point where energy ends and mass starts", while that's clearly subjective. And the zeroth law (temperature is a transitive (comparative) relationship) is just a way of formalizing... a feeling. Yes, temperature is a feeling, there are no hot atoms or molecules. Then, thermodynamics is the scientific description of a feeling ("dynamics of temperature"). Plank died (and accidentally solved the essential problem in quantum physics) trying to prove that thermodynamics was a fundamental fact of nature, and trying to prove that Boltzmann was wrong. Well, thermodynamics isn't a fundamental fact, so whatever Boltzmann says is irrelevant at a fundamental level.

Kant said that we cannot know the truth (you assume that every theory can be interpreted subjectively, and that's a consequence). Because our interface with the real world are our senses. And so, Kant states that we live in a complete tautology: there's no final rule able to validate other rules. Our senses tell one thing (what Kant called phenomena), and math tells a different thing (noumena, reality, what we don't have access to). But even math is built over such tautology. Perhaps now you can understand why it could be possible that the work of Einstein would be applicable only to the result of our sensations.

Worst even, boltzmann thermodynamics is clear about that (boltzmann addresses THINGS), but few people pay enough attention: macrostates are what we perceive, the phenomenon, and microstates address what seems to happen down there (the noumenon). So, statistical thermodynamics assume the existence of things, by converting massive behaviors into static ("statistical") and constant perceptions. Entropy can be a flat curve only at a statistical level. In reality, it would be similar to the output of a seismometer (of course, at a hugely large resolution).

So, here's my personal conclusion: Science is built over the result of our perception (the phenomenal reality), using math, which can provide us access to the noumenal reality (which is inevitably subjectively-interpreted). But we haven't been careful enough to make a clear difference between the science that depends on our perception (thermodynamics) and the science that takes our subjectivity into account (quantum mechanics, which considers the subject, us, the observers, and the object... well, if a fundamental particle can be named an "object"). Perhaps the incompatibility quantum mechanics/gravity lies there (https://www.space.com/32147-why-is-gravity-so-hard-to-understand.html).

So, any mathematical statement is inevitably subject to interpretations. But there's no final truth: we exist, understand, think and make science upon Kant's tautology. We can create rules to validate other rules (e.g. this is science because it was developed using the scientific method). But there's no final rule that would be able to validate that.

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    +1 I like your answer and feel it makes much sense. But the embargo on knowledge you speak of concerns empirical evidence, which is always theory-laden, and does not apply for all knowledge. You cannot claim there is no final truth or that we cannot know it for this is just an opinion. What you say would be true for the empirical sciences but not for philosophy or non-empirical experiential investigations. Still, it gets a vote from me. – PeterJ Dec 7 '18 at 10:47
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    'Our senses tell one thing (what Kant called phenomena), and math tells a different thing (noumena, reality, what we don't have access to)'. This is not what Kant says; he does not allocate what the senses 'tell' us to the realm of phenomena and what math 'tells' us to the realm of noumena. This is not the phenomenal/ noumenal distinction at all. I don't know of any Kant scholar who would put math on the noumenal side of the distinction. – Geoffrey Thomas Dec 7 '18 at 15:43

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