What are some critiques of my philosophy about approaching claims of truth using the scientific method? [closed]

Question

mathematician here. I occasionally go down philosophy rabbit holes and end up in some dark mental states. It always stems from examining the foundations of mathematics.

As a foreword, I am not making dogmatic claims, I am simply stating sentences informed by evidence presented to me over the course of my life. I do not believe that anything I say it 100% true, including this statement.

Each time, I come to the conclusion that absolute truth about anything cannot be known. I have seen the Munchhausen Trilemma, and have not seen a satisfactory refutation of it.

My response to this, is to make claims of knowledge using the scientific method. This means we acknowledge whatever claim of knowledge is subject to further investigation and testing, but it is informed by the most substantial body of evidence available.

You might say, that we know 1 + 1 = 2 is certainly true. However, this is essentially a definition, following from the definition of the successor and addition operations. It's like asserting the definition of the word "mountain" is true. From evidence available to me, all evidence suggests that definitions are neither true or false.

You might be certain that "you are breathing", but there is evidence suggesting a "self" does not even exist hence it is not "you" breathing, but I will digress.

This is possibly a poor demonstration of my philosophy, but I'm essentially saying one should always be open to the possibility of refutations and criticisms of their beliefs and claims presented with contrary evidence. Any elaboration here would be appreciated.

A more meaningful question, would be something like: is every real number is either less than 0 or greater than or equal to 0?

Let me demonstrate my scientific approach to ascertain the "truth" of this mathematical proposition:

There are lots of mathematical proofs, which can be considered as evidence supporting this claim. Furthermore, one can say that within a system of classical logic and ZFC, this is indeed a true statement, according to the definition of mathematical truth. These arguments show the claim logically follows from a few assumed statements.

Hence we will investigate these assumptions. In classical logic, we assume the Law of Excluded middle is indeed valid, allowing for non-constructive proofs such as proof by contradiction. The non-logical axioms are quite intuitive.

There is no constructive proof of this claim. Furthermore, there is some evidence suggesting that real numbers do not actually exist. However, this is not substantial evidence.

To investigate this mathematical proposition further, we should investigate non-constructive proofs and classical logic.

Considering the applications of classical mathematics within the real world, there is alot of indirect evidence supporting non-constructive proofs. They rely on the law of excluded middle. We will proceed with an investigation of this logical proposition.

Even though computer programs operate within a finite domain, the law of excluded middle is used to evaluate propositions with great effectiveness (if - then statements, see church-turing thesis). In computer programs, all numbers are rational (no space for infinite expansions), we can still verify that each quantified mathematical statement is either true or false according to the definitions of operations and variables involved within the well-formed formula. This counts as direct evidence supporting the validity of the law of excluded middle since any finite mathematical system can be modelled by a computer program.

Applying scientific reasoning via the principle of induction, the extension of this restricted version of mathematics to the infinite domain of classical mathematics and real numbers seems reasonable (a model of infinite computer programs). As previously established, the unreasonable effectiveness of classical mathematics in the natural sciences and engineering is substantial indirect evidence supporting the validity of this system.

Even though there has been evidence suggesting possible logical inconsistency (Godel Incompleteness), after 100 years of extensive use, no inconsistencies have appeared. There is more evidence suggesting the system is consistent as well, but I won't go into it.

From this evidence, it is reasonable to infer that the validity of a statement like "is every real number is either less than 0 or greater than or equal to 0?" is supported by a very large body of evidence, with very little evidence to suggest the contrary, and so accepting this statement is the most informed decision one can make.

-- I am happy to be criticized / presented with evidence contrary to my claims, it's part of my philosophy and scientific inquiry.

Answer 1

You might say, that we know 1 + 1 = 2 is certainly true. However, this is essentially a definition, following from the definition of the successor and addition operations.

Given certain definitions (e.g. the Peano axioms), 1 + 1 = 2 is true. Given other definitions (e.g. "+ means string concatenation"), 1 + 1 = 2 is false.

It's like asserting the definition of the word "mountain" is true.

Not really. Definitions are neither true nor false: they just signify what we're talking about. Some concepts are more useful than others: "mountain" refers to those really big rocky hills that make up mountain range, which is a pretty useful concept, but "a member of {3, 5, ⅙π²}" is not very useful at all. It'd be fairly wrong to designate a word (e.g. splonch) for this concept (and very wrong to say that "even number" means this concept, because that'd be all sorts of confusing).

Furthermore, there is some evidence suggesting that real numbers do not actually exist. However, this is not substantial evidence.

You're making a category error. When people say "the real numbers exist", they mean things like "aspects of reality behave in a way that can be accurately and precisely predicted with descriptions that involve the concept of real numbers". Nobody thinks you can put π in a glass cabinet and display it in a museum.

There is no evidence that real numbers do not "actually exist", because "actually exist", in this sense, is not meaningful. There's no evidence they do "actually exist", for the same reason.

In computer programs, all numbers are rational

Are they? Computer programs can perform symbolic computation, just as easily as humans can.

Applying scientific reasoning via the principle of induction, the extension of this restricted version of mathematics to the infinite domain of classical mathematics and real numbers seems reasonable

Not all properties of finites are properties of infinites. (For example, all totally-ordered finite sets contain a supremum, but not all totally-ordered infinite sets do.) None of your reasoning in these paragraphs is particularly scientific, nor does it stand up to much scrutiny. (There's little point going into the details because your main error was earlier.)

It seems like these arguments were written to fend off an existential crisis, rather than because you think they're actually valid arguments. I expect you've been going through a pattern of doubting, panicking, justifying, then gradually seeing the holes in the justification.

As far as I can tell, the main thing you're missing is this: reason alone cannot tell you what world you are living in.

You can think of different mathematical foundations as different parallel "mathematical worlds". In ZFC, a pea can be chopped up and reassembled into the sun, but in ZF+AD, every set of reals is almost open. In both worlds, 2 + 2 = 4.

Of course, we don't live in any of these worlds. We live in the real world, of things that exist. (Even if Tegmark's mathematical universe hypothesis is, in some sense, true, we live in the world of rocks and apples and clouds and fish, and the world of words and memes and song, and – occasionally – the world of molecules and quarks, or the world of viruses and protists and DNA, or the world of novas and galaxies and black holes.)

We're not looking down on this world from outside: we're part of it. All we can see is how it is now, but we don't necessarily know how it will be, or how it was. The world contains patterns, and we've picked up on many of those patterns: vision appears to correspond to objects, objects appear to exist even if you're not looking at them, etc.. But these aren't statements about the world: they're statements about a model of the world.

Perhaps everything outside your field of view is densely-packed Hitlers: how would you know? But, then, if only things you can't observe are densely-packed Hitlers, and everything else contains what it should, those Hitlers will have no effect on your life. The world in which you live contains no densely-packed Hitlers, and it matters not what "actually exists".

(Readers: if a non-comedian philosopher has proposed this thought experiment, please let me know. "Densely-packed Hitlers" is both insulting and undignified.)

The real world is that which determines what you observe. Do numbers interact with you? Well… you can imagine them. Is that observation? Well… depends on what "observation" means. (I think you might like later Wittgenstein's take on this, though Hilary Putnam's also got a lot to say in Reason, Truth and History.)

Here, we run into another issue. We tend to think thoughts like "trees exist" or "heaps exist", and then get all upset when the Sorites paradox gets proposed. Our ideas and concepts are models, approximations: we live in an illusionary reality, and for all it might be similar to the real world, it might always differ. (Given what we know about the relative sizes of human brains and planets, my best guess is that it will always differ… but I can't know that for certain.) For this, you might like The Open Society and Its Enemies by Karl Popper.

If this fails to improve your mental state, you might like some comics about existentialism.

Answer 2

Even though there has been evidence suggesting possible logical inconsistency (Godel Incompleteness), after 100 years of extensive use, no inconsistencies have appeared.

This is simply wrong. The incompleteness theorems do not suggest any logical inconsistency at all. You cannot conflate truth and provability.

There is absolutely zero evidence that there is any problem with classical FOL, and there is also overwhelming empirical evidence that (first-order) PA is meaningful and has real-world meaning. Beyond PA, and the conservative extension ACA0, it is unclear what (if any) mathematics is needed for real-world applications. Perhaps up to ATR0 or ATR, but so far all concrete applied mathematics has been shown to be provable within ACA0. For reference look up reverse mathematics.

Answer 3

Each time, I come to the conclusion that absolute truth about anything cannot be known. I have seen the Munchhausen Trilemma, and have not seen a satisfactory refutation of it.

isn't the trilemma a good candidate for absolute truth?

My response to this, is to make claims of knowledge using the scientific method. This means we acknowledge whatever claim of knowledge is subject to further investigation and testing, but it is informed by the most substantial body of evidence available.

you realise that what counts as methodology, experiment, evidence, etc., is not god-given, right? that it is, 'at the bottom', the scientific method itself is subject to convention, tradition, opinion, etc.,... right?

From evidence available to me, all evidence suggests that definitions are neither true or false.

yep, definitions are always eliminable, leaving only primitive terms/notions

but there is evidence suggesting a "self" does not even exist hence it is not "you" breathing

i'm curious, which evidence is this?

Furthermore, there is some evidence suggesting that real numbers do not actually exist.

what do you mean? aren't some irrationals computable? aren't there conservativity results for some fragments of second order arithmetic over first order? what about decidability of real closed fields??

Considering the applications of classical mathematics within the real world, there is alot of indirect evidence supporting non-constructive proofs.

if weak/finitary theories suffice for developing the empirical sciences, as some people - Feferman, Ye - argue, and the stronger theories in turn are not conservative, then it's not clear that applicability should count as evidence

This counts as direct evidence supporting the validity of the law of excluded middle since any finite mathematical system can be modelled by a computer program.

isn't this just the fact that LEM simply holds in finite contexts?

Applying scientific reasoning via the principle of induction, the extension of this restricted version of mathematics to the infinite domain of classical mathematics and real numbers seems reasonable

it's not clear that this is reasonable: the empirical sciences never have had to deal with actual infinities, so the analogy is not so good, plus the fact that ω behaves exactly like an inaccessible cardinal is/should be scary

What are some critiques of my philosophy about approaching claims of truth using the scientific method?

for mathematics at least, i guess it's better that we stick to proofs, and use truth as a tool

Answer 4

I must admit I did not understand what "your philosophy" is.

But you are not alone. Many if not all of the philosophical terms you are using do not have a single accepted definition, and their pages on Wikipedia or SEP are long and convoluted. All of them are considered "hard" problems with no clear solutions nor even very clear definitions:

• Truth
• Knowledge
• "Self" (i.e. dualism vs. non-dualism)
• Reality overall
• Reality of concepts, specifically (i.e., numbers)

Some specifics:

• The term or concept "evidence" does not apply to philosophy or maths. "Evidence" is used in the scientific method, to falsify predictions made by theories about the real world through objective measurement. Nothing from objective reality can be evidence for philosophy or maths, and nothing from philosophy or maths can be taken as evidence for anything outside of their respective field. They can "fit" or make useful predictions or give useful explanations (via other sciences using applied maths), but nothing more or less.
• Mathematics do not really fit in your lamentations about Truth, Knowledge etc.. Mathematics can be viewed as a building made up from axioms plus logic. You can then think long and hard about what it all means, but viewing mathematics as outside of objective reality is nothing far-fetched and makes everything much easier (but still very hard). The philosophical concept of "truth" does not apply to mathematics - yes, a statement can be true or false, but this is not the same kind of capital-T-Truth as what philosophers are talking about.
• One sub-topic of this: it is very much not known whether the universe is continuous or discrete at the very lowest levels. I.e., it is unknown whether the full "resolution" of the real numbers from maths is actually needed, or whether any two of the very closest positions in space or time are actually always a certain minimum discrete distance away from each other.
• Science does not have the concept of "truth". Science has falsifiable theories that are considered until disproven (through objective measurement), but none of the not-yet-falsified theories can be considered "true". There cannot, by definition, ever be anything regarded as "true" in science, and this is very much by design. It is, overall, simply the best explanation we have for whatever we are witnessing around us, at every given point in time.