Why? Modern math (and logic) explicitly don't make any absolute claims. They are all just "if you take these axioms as given, then you can conclude these theorems."
Math doesn't care where you get your axioms from, or why you take them as given. So faith is just as good or bad a source as intuition or empirical evidence as far as logic is concerned.
I'm not religious myself, but compatibility with logic is a really, really low bar to clear for any Weltanschauung.
(And you can believe that Knuth and the author of the paper are smart enough to be able to reason away any inconsistencies.)
To be honest, I struggle to explain this to mathematicians even: many of them do not grasp the fact that axioms are basically the things you put your faith in.
I only claim that the main difference between science and religion, which are both based on faith in unprovable statements, is that science invites you to challenge those statements (easy example is the axiom of parallelism), whereas religion doesn't.
In some sense, mathematics is just a game you play. Playing round with different axioms is encouraged.
Another view at axioms is that they are like an API:
Whenever you find a system (real world or elsewhere in math) that fits the axioms of group theory, you can apply all kinds of already proven theorems to it. Very much like building a library on top of an API spec.
In this perspective, faith doesn't come into the picture at all: the axioms of a specific theory are just the shape of the socket that you need to fit your plug in to get already proven theorems for free.
(A somewhat similar special case is reductions in computer science: where eg you show that some new problem is at least as hard as one we already know to be NP complete.)
Sure, that speaks to the usefulness of mathematics and the way it's constructed: perhaps the API analogy is the way to go about explaining it to others.
However, I am mostly talking about people (mathematicians even) who do not understand that something as basic as 2+2=4 relies on our faith that: 1. there exists "one", and 2. there exists a successor to "one" (simplified, for interested readers, look for Peano axioms). The beauty of mathematics and the human mind are in that everything else flows from there!
Most of them would claim that this is universally true, whereas as a mathematician, one needs to understand the difference, and be open to, as you put it, playing with assumptions!
> However, I am mostly talking about people (mathematicians even) who do not understand that something as basic as 2+2=4 relies on our faith that: 1. there exists "one", and 2. there exists a successor to "one" (simplified, for interested readers, look for Peano axioms). The beauty of mathematics and the human mind are in that everything else flows from there!
I don't see any faith in here:
Math just says 'if something like 1 were to exist, and if successors were to exists (etc), then after a long chain of reasoning you could conclude that 2+2=4'.
The 'faith' perhaps comes in when you go from '2+2=4' to eg two hens in my coop and two more hens in my coop means that I have four hens in my coop.
(And that's not trivial! Not all things in the real world or even in math behave like that. For some not even as an approximation.)
You are getting caught up in what mathematics does say: and you are right about that.
I am contrasting that with what people think it says (i.e. 2+2 is 4), without understanding that there's an "if" in there. They take it for granted as indisputable truth, meaning that they have faith in those "ifs" being fulfilled.
Empirical observation is not considered much proof of abstract principles because of the measuring errors.
I.e. Newtonian mechanics is empiricaly observed to be true with a certain error margin during measurements. As you lower measurement errors, you start to notice things that are not consistent with it.
Newtonian physics was and still is very much applicable for many use cases, just like mathematics is.
You seem set on "defending" mathematics where nobody is attacking it.
My comment is not to discredit mathematics: it is the ultimate expression of the human mind, showing the limits of our comprehension. Limits that clearly show that we can start reasoning only if we accept a few things as a given (what I simplified to "faith").
It is beautiful and transcendent in its expression, and while people consider it the most complex of human expression, I consider it the simplest full expression, worked down to the smallest ambiguities our brain will let us have. It is a reflection of what our mind can understand and grasp, and we should accept it as such so we can both grow our understanding of the world, and test limits of our understanding.
b) Religions don’t just make unprovable claims, they make contradictory claims. Mathematicians prefer systems which are consistent yet have unprovable claims as opposed to inconsistent systems.
The idea of a language which underlies the structure of the natural world (the word / logos) is a core underpinning of -- I hesitate to say most, but at the very least -- many sophisticated theologies.
The line between that which you can call "the word" and what you call "math" is not clear to me at all.
That's true, but most people people consider mathematical truths to be actual truths, and this is backed by the predictive power of mathematical models.
You can doubt the predictive power of math itself, but I have yet to see a clear example where rigorous math deviates from nature.
The fact that effects seem to follow causes naturally leads to the idea that there is a language which you can use to symbolically represent the process by which the universe evolves in time and space. The attempt to realize that language is what we call "math."
My point is only that the idea of the logos is not at all dissimilar from that, and the idea that someone who is intensely interested in logic and mathematics would also be interested in theology is only natural to me.
While I myself am a strident atheist, I must point out that your comment demonstrates a profound ignorance of the history of both faith and logic. In the premodern era it was through logic that the faithful intellectuals discovered “the mind of god.”
as someone who spent their 20s and 30s devoutly atheist I very much appreciate the sentiment - never discount religious people. the smartest person you meet in your life will be religious, probably.
“A 1998 survey based on a self-selected sample of biological and physical scientists of the National Academy of Sciences in the United States found that 7% believed in the existence of God, 72.2% did not, and 20.8% were agnostic or had doubts.[53] Eugenie Scott argued that there are methodological issues in the study, including ambiguity in the questions. A study on leading scientists in the US, with clearer wording and allowing for a broader concept of "god", concluded that 40% of prominent scientists believe in god.”
Right and we created god too! I think that atheistic people get too bent up when someone thanks god for something. You could equally say “thanks for all that exists and the way that things turned out” but it’s a mouthful!
You can redefine God to mean 'all that exists and the way that things turned out'. But then you don't get to turn around and use the definition of God that goes 'whatever it says in my preferred holy text'.
