Is Math Certain? Biochemistry Senior Seeks Answer

[OchemAndy]

Hey I am new to this and currently am a senior undergraduate majoring in biochemistry. Earlier today I got into a sort of debate about the certainty of math. I believe it to be true but is it set in stone?

I wanted to ask; how certain is math? More specifically how certain is arithmetic? I realize that the basic elements of math are made of axioms which help build proofs that further support mathematics. Logically I think that makes some of the more basic math 100% certain especially arithmetic. Is there anyway arithmetic could get overturned in the future by means of future discoveries?

I was wondering if anyone could give me a dumb-ed down version of this answer?
Thanks

 

[Simon Bridge]

Welcome to PF;
The answer depends on what you mean by "certain" and what you mean by "true". You need to define your terms.
1+1 is always 2 by definition of what the symbols mean.

It is possible to misapply the symbols ... ie, one cup of water added to one cup of popcorn does not get you two cups of soggy popcorn, but that would involve a false equivalence fallacy.
But the fallacy is only a problem if logic is true right?

How do you decide if logic and maths is true without using logic or maths?

In philosophy, the rules of logic and arithmetic are taken as properly basic.

That is different from saying they are true ... it does mean that it doesn't matter.

They form a language that we use to describe everything else. These are the rules that we use to define what we mean by "truth" so asking if they are true themselves is to make a category error, arguing that they are true is circular reasoning, and the assertion that they are not contains a logical contradiction. The best you can do is say it is internally consistent.

(Then there are the situations like in logic: "This statement is false.")

But this is a physics forum.
In physics, something can also be true to the extent that it is congruent with reality. We cannot say that maths is true in this way, but we can say that it is useful for discovering truths ... using maths is a reliable way to find out what other things are true.

Not everything that is mathematically true is also true in fact.

Extending... we have good reason to suspect that the rules of maths and logic arise from the physical laws of the Universe, to our brains, via evolution. However, we do not have a causal link so we do not know this for a fact. It is common to have to make do with imperfect knowledge.

Bottom line: since strict truth is defined by maths, then the statement that maths is not true would include a contradiction. This sort of thing comes up when a conversational use of a word is being used in a technical context. Can you come up with a definition of "true" that does not involve maths directly or by implication?

Note: going in-depth in the philosophy of these things is, iirc, banned in these forums.
So that is as far as I will go. Look up "properly basic" for more... also "empiricism".

 

[mfb]

How do you define certainty?

The proofs in mathematics all follow from the axioms and definitions. Unless some mathematician made a mistake in the proof and no one spotted it (not impossible, but very rare), a proof is absolutely correct. The relevance for our universe, other sciences and so on is a different question.

 

[fresh_42]

There is no need to push this question to philosophy. The question whether arithmetic axioms are free of contradictions is Hilbert's second problem on his famous list (1900) which hasn't been answered until Gödel 1931. So I wouldn't call three decades of mathematical research philosophy.

 

[jedishrfu]

Here's a good video on the crisis of math that occurred when mathematicians attempted to base all math on set theory as presented by Marcus du Sautoy:



More on Godel's theorem:

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

 

[mathwonk]

math conclusions re exactly as certain as their hypotheses, and the logical validity of the arguments. there is no free lunch, i.e. mathematicians also make mistakes, both from carelessness, haste, and misunderstanding. as time passes, these flaws are improved.

 

[symbolipoint]

math conclusions re exactly as certain as their hypotheses, and the logical validity of the arguments. there is no free lunch, i.e. mathematicians also make mistakes, both from carelessness, haste, and misunderstanding. as time passes, these flaws are improved.

Yes, Mathematicians and the rest of us can use improved flaws. The world can always use better flaws.

 

[Svein]

Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess.
-- Robert A. Heinlein --

 

[PeroK]

Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess.
-- Robert A. Heinlein --

It's remarkable the nonsense some people come up with to try to be clever.

 

[Svein]

It's remarkable the nonsense some people come up with to try to be clever.

Well, it is aptly observed. I am a mathematician and I am well aware that we are tool-makers for other sciences. But we are also aware of the difference between "das Ding als sich" and "das Ding für mich" to quote a famous philosopher - we know very well that the mathematical description scientists come up with is just a description, not the phenomenon itself. The problem is that sometimes scientists fall in love with the description and refuse to accept that the description sometimes does not fit the "real world". To quote a sergeant: "In this army, if the map and the terrain do not match, the terrain is always right".

 

[PeroK]

Well, it is aptly observed. I am a mathematician and I am well aware that we are tool-makers for other sciences. But we are also aware of the difference between "das Ding als sich" and "das Ding für mich" to quote a famous philosopher - we know very well that the mathematical description scientists come up with is just a description, not the phenomenon itself. The problem is that sometimes scientists fall in love with the description and refuse to accept that the description sometimes does not fit the "real world". To quote a sergeant: "In this army, if the map and the terrain do not match, the terrain is always right".

I can see no relation between this and what Heinlein said. You're trying to defend the indefensible.

 

[Svein]

I can see no relation between this and what Heinlein said. You're trying to defend the indefensible.

Then we are looking at it from very different viewpoints. Sorry.

 

[symbolipoint]

This current disagreement resembles the relation between "descriptive" and "prescriptive".

 

[sysprog]

Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess.
-- Robert A. Heinlein --

It's remarkable the nonsense some people come up with to try to be clever.

Well, it is aptly observed. I am a mathematician and I am well aware that we are tool-makers for other sciences. But we are also aware of the difference between "das Ding als sich" and "das Ding für mich" to quote a famous philosopher - we know very well that the mathematical description scientists come up with is just a description, not the phenomenon itself. The problem is that sometimes scientists fall in love with the description and refuse to accept that the description sometimes does not fit the "real world". To quote a sergeant: "In this army, if the map and the terrain do not match, the terrain is always right".

I can see no relation between this and what Heinlein said. You're trying to defend the indefensible.

Then we are looking at it from very different viewpoints. Sorry.

The epistemic distinction between the phenomenal (Kant: "das Ding für mich") and the noumenal (Kant: "das Ding als sich") maps to the set-theoretic distinction between elements within sets and real-world objects that such elements may be defined or postulated or imputed to represent.

 

[fresh_42]

"das Ding als sich"

It has to be "Das Ding an sich".

 

[nuuskur]

The thing that drew me into mathematics is once a sound proof has been presented, there will be no more debate. A correct proof is always stronger than any counterargument. This removes the compromise of appealing to ignorance or arguments ad populum.
Of course, we do rely on axioms - statements we regard to be true without proof. We also rely on mathematical logic. One could question the consistence of either component. That, however, is an exercise in the foundations of mathematics, veering somewhere between mathematics and philosophy.

When you talk about arithmetic and whether a result could change, it depends. For instance, you argue that it is always true that $1+1=2$. By default you are equipped with the arithmetic of natural numbers. I could say that $\vert \Delta \vert \sim \vert\vert$ is also always true. Well, you're not sure anymore, are you? So, why is $1+1=2$ set in stone, then? It isn't!

What does $+$ mean? What are symbols $1,2$? What does it mean to be always true to begin with.
Welcome to the world of predicate logic :)

 

[Svein]

It has to be "Das Ding an sich".

Yes. In my defense - my introduction to philosophy was in 1963.

 

[Svein]

Of course, we do rely on axioms - statements we regard to be true without proof.

Nitpicking - axioms do not have to be "true" (whatever that may mean). They have to be "useful" in some sense, though.

 

[PeroK]

Nitpicking - axioms do not have to be "true" (whatever that may mean). They have to be "useful" in some sense, though.

@nuuskur did not say axioms are true, he said we "regard them to be true", which is precisely what we do, if we allow true to mean true within our mathematical system.

axioms - statements we regard to be true without proof

On the other hand, there is no requirement that an axiom be "useful", which is an undefined term as far as I am aware.

 

[fresh_42]

The charming point with axioms is, that they usually reflect statements people don't question, e.g. first order logic. If they are questioned, one can find everyday experiences which would turn absurd. This is not a necessity of an axiomatic system, but a property of those we usually work with. As a consequence, they provide a common ground for a discussion and the mechanisms @nuuskur described apply: No strange exits anymore.

I have always been fascinated by the standard proof for the existence of infinitely many primes. It's so easy, that you can tell a kid in elementary school. It makes the difference between calculation and mathematics. It's a pity that schools mainly still teach calculation only. It's difficult to see what a patrtial fraction decomposition is good for in real life, and far easier to see what it is good for in the world of mathematics and physics. And I do not accept "non scholae sed vitae discimus" because I had to learn everything about cells in biology and haven't ever needed it either.

 

[PeroK]

There are many examples of where mathematics appears to underpin the real world. One that I particularly like is:

Take a regular book. It has three dimensions. The one I have chosen has a thickness of about $2cm$, a width of about $16cm$ and a height of about $24cm$. Appropriately enough, it is "Analysis: The Theory of Calculus". There are three primary axes about which you can spin the book. The experiment is to spin the book about these axes and let it fall. Best to keep the book closed with an elastic band first.

If you spin it about the axis going through the shortest dimension (thickness) or an axis along its longest dimension (height) it keep spinning as it falls.

If, however, you spin it about the axis going through the middle dimension, it falls chaotically and doesn't spin in the direction you started it.

When you analyse the rigid body motion, the differential equation that results has a negative sign in the first two cases, resulting in a harmonic (stable) solution. And, the equation has the opposite sign in the third case, resulting in an exponential (unstable) solution.

To me, this demonstrates the underlying mathematical nature of reality. I can't imagine explaining this, let alone predicting it before you see it, without the use of mathematics. Wherever the human mind got it from, the mainstream mathematics that we all learn is fundamentally woven into the natural world.

 

[SSequence]

The charming point with axioms is, that they usually reflect statements people don't question, e.g. first order logic. If they are questioned, one can find everyday experiences which would turn absurd. This is not a necessity of an axiomatic system, but a property of those we usually work with.

In real world we usually our domain of discourse is based upon finite number of objects. In that case the logical equivalence of say something like $\forall x \, [p(x)]$ and $\neg \exists \, [\neg p(x)]$ should be taken beyond any reasonable doubt.

However, in mathematics, it seems to me that we are often talking about some infinite collection of objects. The main difference (that I feel) is that there are two factors:
(1) Does LEM apply?
(2) Is the domain of discourse surveyable?

Now considering the domain of number-theoretic assertions. It seems to me that (2) definitely be correct since we are running over natural numbers. I don't know about (1) for sure but even if it is true, I find it a very important and highly non-trivial assumption ... even if my personal inclination is towards it being correct. I have repeated this too many times on this forum (along with more detailed reasons few times) ... so moving on to other points.

Taking (1) and (2) to be both correct, it seems to me that much of our usual ways of thinking should be correct and, in that case, it seems to me that PA should be sound (since I am not comforable with the specifics, I am wording it cautiously).

Taking just (2) to be correct for sure (which it is), it seems to me that con(PA) should be taken as a real problem. But I read somewhere roughly along the lines of, "if one accepts the given well-order of $\epsilon_0$ (in Gentzen's proof) then it is nearly self-evident that PA is consistent". I hardly know anything about the proof but if that is true, then it seems to me that con(PA) should be taken as a settled problem (in positive) ... ofc giving some room for human error (which is "always" a possibility anyway ... even for simple arithmetic ... esp. with very large numbers).

Regarding point(2) in a more general sense, see the second section ("surveyable concepts") in this essay: https://arxiv.org/pdf/1112.6124.pdf. I linked this because I don't know of any other author mentioning or discussing this particular idea in detail. However, I don't know what is the "limit" of the notion of "surveyable" that the author has in mind. At any rate, this certainly seems to be an important concept.
This is relevant in the specific discussion because it is certainly not clear at all to me why every reasonably formulated statement (even generously assuming that we can answer all reasonable questions about surveyable domains) about an unsurveyable domain should have a definite truth value (and this would probably give rise to being very careful about corresponding rules of inference when dealing with the given unsurveyable domain).

===========================

The shorter answer to OP's question is that many people seem to believe that number-theoretic assertions have definitive content and truth value (though there might not be definitive agreement on the means of arriving at truth). On the other hand, some people definitively seem to doubt the "meaningful content" w.r.t. statements that involve set theory.

In particular, I like the following quote*** (Feferman on the Indefiniteness of CH --- Peter Koellner):
"In what follows I am will exposit and extend Feferman’s critique, argue that each component fails, and conclude that when the dust settles the entire case rests on the claim that the concept of natural number is clear while the concept of arbitrary sets of natural numbers is not clear."

I do not agree or disagree that such a conception is not clear (in principle), but this certainly seems to be an important problem (at least from my naive pov ... as I have mentioned, perhaps too many times, on this forum).

*** Of course I don't understand anything at all about the article :p

 

[Svein]

To me, this demonstrates the underlying mathematical nature of reality. I can't imagine explaining this, let alone predicting it before you see it, without the use of mathematics. Wherever the human mind got it from, the mainstream mathematics that we all learn is fundamentally woven into the natural world.

To me, this demonstrates that the mathematical model of rigid-body dynamics are correct in the eyes of an observer.

What we do not know is whether the model is accurate at astronomical sizes and/or extremely high rotational velocity...

 

[rude man]

Hey I am new to this and currently am a senior undergraduate majoring in biochemistry. Earlier today I got into a sort of debate about the certainty of math. I believe it to be true but is it set in stone?

I wanted to ask; how certain is math? More specifically how certain is arithmetic? I realize that the basic elements of math are made of axioms which help build proofs that further support mathematics. Logically I think that makes some of the more basic math 100% certain especially arithmetic. Is there anyway arithmetic could get overturned in the future by means of future discoveries?

I was wondering if anyone could give me a dumb-ed down version of this answer?
Thanks

One does wonder.
In his "Principia Mathematica" Bertrand Russell took over 1000 logical steps to prove that 1 + 1 = 2. That's the good news.

The bad news is that mathematician Kurt Goedel was able to refute the whole work!

 

[Dr Whom]

The simple answer is yes.
More specifically it is a self consistent set of axioms from with everything else follows.
Finally, certain mathematical systems correspond to a greater or lesser degrees to systems in the universe. For example, Newton's laws of motion are wrong but so accurate that they are extremely useful. They are approximations of Einstein's special relativity, that is as accurate as we have been measure but only needed where Newton's break down. It may be that at some point these will prove to be only an approximation of some yet more accurate theory.

 

[SSequence]

One does wonder.
In his "Principia Mathematica" Bertrand Russell took over 1000 logical steps to prove that 1 + 1 = 2. That's the good news.

The bad news is that mathematician Kurt Goedel was able to refute the whole work!

What does refute mean in that case? Generally speaking, if one does adopts consistency as the minimal condition, then the failure of consistency can be thought of as refutation in some sense.

But the second incompleteness theorem just shows that consistency of a (consistent) system can't be proved by the system itself.
However, it doesn't say anything more than that. It doesn't even rule out a proof of consistency from outside means (Gerhard Gentzen's proof and probably later results along similar lines etc.)

 

[lavinia]

Well, it is aptly observed. I am a mathematician and I am well aware that we are tool-makers for other sciences. But we are also aware of the difference between "das Ding als sich" and "das Ding für mich" to quote a famous philosopher - we know very well that the mathematical description scientists come up with is just a description, not the phenomenon itself. The problem is that sometimes scientists fall in love with the description and refuse to accept that the description sometimes does not fit the "real world". To quote a sergeant: "In this army, if the map and the terrain do not match, the terrain is always right".

Physics is also just a description as are all sciences, all pseudo sciences, all arts, all conscious experience. But I wonder what the "phenomenon itself" actually is or even if there is such a thing.

 

[coolul007]

All things are true, until you find a counter example.

 

[lavinia]

All things are true, until you find a counter example.

Before you give a counter example would you say that the statement "All integers are even." is true?

 

[Hypercube]

All things are true, until you find a counter example.

This is like "innocent until proven guilty" - applied to mathematics. However, benefit of a doubt is something we're obliged to give to people, not mathematical statements.

 

[fresh_42]

Before you give a counter example would you say that the statement "All integers are even." is true?

My first thought about a counter argument was an epimorphism and a short exact sequence, not a counterexample.

 

[PeroK]

All things are true, until you find a counter example.

Fermat's last theorem?

Riemann hypothesis?

 

[Svein]

Fermat's last theorem?

Riemann hypothesis?

Collatz' conjecture?

My favorite: A closed form expression for ζ(3)?

 

[fresh_42]

I think the thread has run its course. There is little left to say, which could be even close to substantial.
It is mathematically certain that the uncertainty principle holds.

Thread closed.