HN Books @HNBooksMonth

The proof is not a social construct.

The truth is.

The proof is a mechanism to reach that consensus, by convincing other mathematicians of a specific truth. That is all it is.

There is a naive idea that a proof is a purely mechanical series of steps that provides access to truth. Last I checked, this isn't so for the vast majority of proofs in math. Such a proof would be way too tedious to construct or check by mathematicians. And if it isn't checkable, how do we know it is actually true?

Automated proofs are a subfield, and (again, last I checked) controversial because they can often not be checked by humans.

So for example, if the proof doesn't convince other mathematicians, then it's not a a proof.

Or it might convince other mathematicians and later turn out to be wrong after all.

For more on the practical aspects of math, I highly recommend The Mathematical Experience.

https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav...

I read it in German:

https://www.amazon.com/Erfahrung-Mathematik-German-P-J-Davis...

⬐ AnonCoward42

> The proof is not a social construct.

> The truth is.

Yeah, that is how it feels like nowadays, however the truth is bound in a narrow set of assumptions. These assumptions are bound in reality even in mathematics (One apple is one apple, you add another one, you have two). And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct.

The details of how a truth is communicated is in a sense a social contruct, because communication as a whole is, however nobody would call it like that. It is maybe a small reminder that meddling with language for no apparent reason is a warning sign, but this is going a bit off-topic.

⬐ mpweiher

[Mathematical truth as a social construct]

> Yeah, that is how it feels like nowadays,

It's always been that way. (Again, I really recommend the book[1] ). And it's hard to see how it could be otherwise.

(Also depends a little about what exact mathematical truth we are talking about and whether you are a Platonist or Constructionist)

That doesn't imply what either the recent proponents or the critics seem to think. It does not at all imply arbitrariness or that anything goes.

> however the truth is bound in a narrow set of assumptions.

Yes, it is. Again, something being a social construct does not make it a free-for-all. More the opposite, because the constraints are socially enforced.

> And while there is an epistemologic level to reality, you would dismiss reality entirely by calling it a social construct.

Mathematics ≠ Reality. Science is about reality, but scientific truth is also a social construct (see Popper), and highly constrained by reality (ibid).

[1] https://www.amazon.de/Mathematical-Experience-Phillip-J-Davi...

⬐ naasking

> Mathematics ≠ Reality

Conjecture. A mathematical universe is consistent with everything we know, in which case math is literally the study of reality.

⬐ mpweiher

This is simply false, for at least three reasons. There are probably more.

1. Math can describe lots of universes that differ from observation. For starters, just look at all the different kinds of space-time there could be.

2. On top of that, even the math that we have that describes our current universe is...tricky: the math at the core of our two best physical theories (General Relativity and Quantum Mechanics) is inconsistent, i.e. they both cannot be true at the same time.

Lots of our greatest minds have tried to find a mathematical formulation that makes the two compatible, but so far none have succeeded. So there may not actually be a mathematical formulation.

3. And finally, nature is under no obligation to be describable mathematically. That it is so describable is a fortuitous circumstance, and one we might be finding the limits of (see (2)).

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

⬐ naasking

> Math can describe lots of universes that differ from observation.

This has all been hashed out already. Look up the mathematical universe hypothesis and Wolfram's Ruliad for two different but similar approaches. The basic idea is that all consistent mathematical objects exist, and we simply inhabit one of them.

The fact that QM and GR are inconsistent with each other is not an issue because they describe different universes in mathematical space that are close but not equal to our own.

> And finally, nature is under no obligation to be describable mathematically. That it is so describable is a fortuitous circumstance

That too is a conjecture. There is exactly zero evidence that nature is non-mathematical, but people who keep saying this are basically insisting that it's effectiveness at describing reality is simply a continuous chain of grand coincidences. To say this is statistically implausible bordering on impossible is the kindest way to put it.

⬐ feanaro

Saying Mathematics ≠ Reality fails to capture a large portion of the story, since a subset of mathematics is clearly necessary to be able to encode science, and confirmed by science, and is in that sense a part of reality.

⬐ mpweiher

It's not necessary. It is empirically useful.

Mathematics is about describing possible worlds. Given these assumptions (including the rules of the game), what follows?

Science is about figuring out the real world. The real world has no obligation to be describable by mathematics. That it is so describable is fortuitous.

"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

⬐ feanaro

> It's not necessary.

I didn't mean "necessary" in the sense of modal logic, but in a more colloquial sense. Physical laws as known to us today are described using mathematics. We know of no other viable ways to describe them. In that sense, (a subset of) mathematics is necessary for this purpose.

It was just meant as a restatement of the concept of unreasonable effectiveness of mathematics in a more natural language.

> The real world has no obligation to be describable by mathematics.

There is no obligation, but that it is so in our particular world is undeniable fact.

⬐ prmph

Imagine a universe, where, when you combine two apples (or really anything), you always, through some weird physical process, ended up with three items.

Would that invalidate (or not), the statement that 1 + 1 = 2?

⬐ TheOtherHobbes

An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense.

To a non-human consciousness those features may be uninteresting, irrelevant, or incomprehensible, so they might not see apples at all. But they could see " "s, which we don't even have a concept for, never mind a word. And which we either ignore or possibly don't see at all. (Imagine perceiving complex networked relationships directly instead of having to access them through symbolic models.)

There's no reason why math wouldn't be the same. From experiments we know that cats can't count, but they can distinguish sizes. So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue.

I have a theory this is why Hilbert's Project failed and you always end up with an incompleteness theorem.

You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.

So you can define integers in various more and more obscure ways. But fundamentally you have to start with the subjective experience of "integer" as a concept that matters to you. And you can't prove a subjective experience objectively.

⬐ naasking

> So cat math likely wouldn't have integers as we know them, but would have some kind of size-based analogue.

Sorry, but no. Any species capable of actually creating some kind of math will have some mathematical structure isomorphic to the integers. If cats can't count, then that just says that cats are not capable of creating some kind of math.

⬐ gmfawcett

IDK, it seems easy to imagine an alien mathematics based only upon continuous values? There's nothing obviously universal about discretizing things.

⬐ naasking

Firstly, the reals contain the integers, so there is an isomorphism as I said.

Secondly, discretization absolutely is universal. It's literally in the laws of physics for one (particles are discrete, energy levels are discrete, etc.). For another, are you suggesting a physical alien species will have a continuous number of appendages, or organs, or that their population will somehow be continuous? I frankly don't see how you can possibly escape aliens capable of math developing a notion of basic counting.

⬐ gmfawcett

Eh, you're not explaining universal truths here, you're just anthropomorphizing. Why must it have appendages, organs, or populations? Why presuppose that its conceptual model includes particles at all? What if a vast, hyper-continuous intelligence simply cannot comprehend the concept of being discrete?

⬐ naasking

Firstly, those were just examples of commonly countable structures, even if they're not universal (which is debatable). Discretely countable structures are literally everywhere and fundamentally inescapable, which is why I mentioned physics. I didn't presuppose physics, the discrete structure of physical reality is directly observable, it's not some fiction we made up.

Secondly, what we know must be bound by what we've observed. You can imagine any sort of being you like, but that doesn't make your imagined creature logically coherent or physically realizable.

Any physically realizable intelligence must:

a) Be differentiable from its environment: that means it must have some enclosing boundary separating an inside that's different than an outside.

b) Have internal structure: intelligence by necessity is structured thought. Structured thought entails differentiable physical structure to hold structured thoughts. Such structure by itself is necessarily countable, being made of matter.

⬐ AnonCoward42

> An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense.

It was clearly not about the apple, but about distinct entities with similar features. Now for another being these might not be similar, but something else probably is, disregard of different dimensions, different senses or the likes. As the observed reality for an alien species or actually any other species on this earth is different of course.

> You cannot create an absolute internally consistent mathematics, because foundational axioms depend on subjective experience, not on objective logic.

> And you can't prove a subjective experience objectively.

I think the misconception comes from the fact that you need basic assumptions to build an abstraction. One of the most basic assumptions is that something like a shared reality exists and we're not for example in a virtual world or a dream.

You can happily deny this shared reality, however I would not necessarily encourage you to touch fire (literally and figuratively speaking).

⬐ marcusverus

> An apple is not an apple. An apple is a subjective construct that summarises the distinguishing features of a certain kind of object as it appears to our sense.

This is utter nonsense. An apple IS an apple. If I put one on a table, then obliterate every human being that’s capable of sensing it, the apple is utterly unaffected.

If aliens come down and experience the apple differently than we would have, that doesn’t change the apple one bit.

⬐ ookdatnog

It's not nonsense, they are exactly right.

The world is a sea of particles and energy which behave according to certain patterns (both fundamental laws and emergent behavior). Some of these patterns are pertinent to us, so we name them, giving rise to a category. "Apple" is such a category.

The clump of molecules on the table we denote with the term "apple" doesn't care that our brains have deemed it similar enough to certain other clumps of molecules to be placed in the same category. If all humans cease to exist, the clump of molecules may still be on the table, but there's no one left to consider it part of any category.

If aliens then visit who can't eat the apple and aren't interested in botany, they may simply choose not to distinguish between apples and pears, or apples and any other fruit, or even apples and any other form of organic material. The same clump of molecules is there, but the categories it belongs to have changed.

⬐ marcusverus

> The clump of molecules on the table we denote with the term "apple" doesn't care that our brains have deemed it similar enough to certain other clumps of molecules to be placed in the same category. If all humans cease to exist, the clump of molecules may still be on the table, but there's no one left to consider it part of any category.

Correct. If all humans cease to exist, the human-made abstraction of "the apple" passes away. But an apple--the one on the table--is completely unaffected.

The difference here is that between an abstraction and a concrete instantiation of said abstraction. The abstraction of "the apple" is indeed a subjective construct which would pass away with humanity. But OP said "an apple". "An apple" is a part of the physical universe. It is, like you said, a clump of molecules. Molecules that can be measured objectively.

An apple is no more subjective than the orbital period of the earth or the length of the standard meter.

The Mathematical Experience is not a history of mathematics per se, but there is a lot of history and context in it. Highly recommended.

https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... is a link to Amazon.

I'm not who you are asking, but this book https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav... hits some of the philosophy of math. It's been a long time since I've dipped into it, but IIRC, it puts mathematical discoveries and exploration into more of a cultural and philosophical context. I could see this being sampled for a good crossover course that would cover humanities requirements for STEM, and STEM requirements for humanities.

My experience is that non-mathematicians like to have GED on their coffee tables as a conversation piece. The related book that mathematicians like is https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav.... (High school math is enough to enjoy it, but the more math you have, the more you'll get out it. All of the way up to the PhD level.)

"The Mathematical Experience" goes into this in pretty good depth.

http://www.amazon.com/Mathematical-Experience-Phillip-J-Davi...

⬐ btilly

Upvoted for bringing up The Mathematical Experience in a relevant way.

If I could get everyone to read just one book about mathematics, that book would be The Mathematical Experience. More than any other book, it gives the actual experience of what mathematics is like.

Much of it is accessible to someone who is still in high-school. And yet it has valuable lessons for people who have PhDs in the subject. I know a number of mathematicians who say that it explains why they went into math. And I can say that in its pages I can find explanations of both what caused me to love math, and eventually to leave it.

⬐ calibraxis

For anyone interested, a new book by Reuben Hersh is scheduled for this Christmas: "Loving and Hating Mathematics: Challenging the Myths of Mathematical Life".

http://www.amazon.com/Loving-Hating-Mathematics-Challenging-...

[Edit: I asked why you left math, out of curiosity, but then I realized you might have a blog...]

⬐ btilly

I take it you found http://bentilly.blogspot.com/2009/11/why-i-left-math.html then? :-)

⬐ calibraxis

Yes, exactly. :-)

If you are interested in this topic and you haven't already done so, check out "The Mathematical Experience"

http://www.amazon.com/Mathematical-Experience-Phillip-J-Davi...