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Comment by bheadmaster | Hacker News Reader

The fact that mathematics gives us power to predict events in the real world makes it independent of social consensus.

If everyone in the world believes that 2+2=5, that doesn't make it less true that 2+2=4 - in the sense that I know for sure, if I take throw two rocks on a pile of two rocks, I'll get a pile of four rocks, not five rocks.

I hate this sociologist view that everything depends on the social consensus. Going extreme with it is how you end up in a 1984-esque society:

    Anything could be true. The so-called laws of Nature were nonsense. The law of gravity was nonsense. ’If I wished,’ O’Brien had said, ’I could float off this floor like a soap bubble.’ Winston worked it out. ’If he thinks he floats off the floor, and if I simultaneously think I see him do it, then the thing happens.

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That's actually an example of what OP was talking about. You have defined + as the operator that mimics what piles of rocks do, and defined numbers as counting rocks.

That's only a tiny fraction of what math does. An interesting and useful one, and mathematicians have put a lot of work into studying basic arithmetic. They have expanded out into numerous other forms, some of which turn out to have correspondence to the real world like non-Euclidean geometry.

Others turn out to be completely abstract and are merely curiosities. There are an infinite number of them, each containing truths, almost all of them of no interest. Interest is defined by mathematicians, not physics. Even so it turns out to sometimes be useful, such as the beautiful theorems of prime numbers that drive Internet security centuries after they were invented.

That is what the OP means. You can make up any axioms you want and prove true theorems. But the hard part is convincing other mathematicians to care.

> But the hard part is convincing other mathematicians to care.

My point is that whether other mathematicians care or not is completely irrelevant and doesn't subtract from mathematics' power of predicting phenomena in the real world.

Each and every mathematical theory has to be consistent with basic rules of reality - if nothing else, symbolic manipulation relies on basic arithmetic and set theory. Without symbolic manipulation, you can't even express all those "abstract" mathematics - to say that "abstract" mathematics can not have correspondence to the real world is completely false, because of this basic connection.

Now that I think about it - claiming that "mathematics is a social consensus" is exactly what I'd expect from a mathematics professor - a person whose whole life is isolated from reality, limited to the rigid structure of academia, and whose whole existence depends on other people caring. I doubt there is a single (professional) engineer that would say something like that.

Reading the discussion, I think you're both right. There is a common sense notion of mathematics that exists independent of what people want to believe or agree upon.

There are also ways to philosophize about the nature of things in order to frame math as a human construct. I think both views can be simultaneously correct

You can construct a formal system with any axioms you choose. It is not arbitrary that some of these systems turn out to be useful in modeling the world. But there are other systems that are only dry exercises in symbol manipulation that may be of little use to physics or engineering. Or of course they might end up being super important 100 years later. But in the meantime mathematicians might be interested in them anyway.

> I doubt there is a single (professional) engineer that would say something like that.

I suspect an engineering professor would be even more compelled to highlight how social consensus is the foundation of everything else that follows. You only have to read the surface layers of this or that performance debate to see how it is unfortunately the case. "Performance" is a fluid term that doesn't mean much of anything on its own -- and people arguing that they've juiced another drop of performance juice from this or that application's fruit are often just talking past each other in terms of priorities or perspective.

The symbolic manipulation at the heart of mathematics is a byproduct of language -- another system beholden to social consensus. It's inescapable.

I know nothing of OP, or the professor he talked to, but I think they might be talking about the higher level concepts of math.

For example, set theory. If set theory allowed self reference you could have set R, a set of sets that do not contain themselves. Would R contain R? It can’t, but it can’t not either. It’s self contradictory. The solution, reached through consensus, was to restrict the definition of the set to not allow sets to reference themselves [1].

[1] https://youtube.com/watch?v=HeQX2HjkcNo

The way I learned how to resolve the Russell paradox: Over here on the left in a pile we have the elements we will work with. Now, for the sets, they are made of the elements and are in a separate pile, are over here on the right. Sooo, with this little preliminary step, there is no way to consider the set of all sets that are not elements of themselves.

As I recall, there was a paper by Robert Tarjan where he observed that most paradoxes are from self referencing. Soooo, rule out self referencing and will rule out most paradoxes!

> If everyone in the world believes that 2+2=5, that doesn't make it less true that 2+2=4 - in the sense that I know for sure

But not everyone believes that 2+2=5, isn't it? I mean, if you criterion to separate truth from lies is the reality, then now you are talking about a counterfactual reality without proving that this counterfactual reality is possible. So this your statement is unproven, I'd say it cannot be proved.

I can argue, that if world believes that 2+2=5 then it makes 2+2=4 to be false. Just think about it. Try to imagine a plausible counterfactual reality where people believe that 2+2=5. They probably would believe that succession of numbers goes like this: 1, 2, 3, 5, 4, 6, 7, 8, 9, 10, ... And in such a counterfactual reality it would be very strange to believe, that 2+2=4. You could if you liked, and you could build a mathematics around it, but you would have a lot of problems of communication with others.

In il nome della rosa William of Baskerville tells us that concepts are signs and words are "signs of signs".

In math logic we can define systems of symbols and logical connectives and deductive rules. We can argue for the correctness of a logical system but we can not do it in the same system; we have to go "meta". Similarly a formal system could be a model of something "real", but the correspondence would be beyond mere logic.

You got it backwards. 1984 is possible because it's fundemental that some truths are social constructs.

I would argue that those aren't truths. They need some other term to describe them but truth doesn't really fit the bill. Conflating them with truth leads to weird outcomes in your reasoning.

A more accurate term might be stories rather than truth.

If Winston and O'Brian both believe O'Brian is levitating, that still doesn't make it actually true. "Sanity is not statistical" as Winston says in the book

> that 2+2=4 - in the sense that I know for sure, if I take throw two rocks on a pile of two rocks, I'll get a pile of four rocks, not five rocks.

That depends on if one of the rocks breaks in half as you throw it onto the rock-pile or not. And also if the resulting piece knocked off is large enough to pass your fuzzy and contextual distinction between "rock" and "pebble".

But IMHO, arithmetic such as counting numbers and 2 + 2 = 4 are not part of the natural world. If I have a rock and another rock, I can with minor effort tell them apart: they have different weight, size, shape, density, composition etc. They are each unique individual assemblages of huge numbers of atoms in distinct never-to-be-repeated arrangements. In what way are these 2 unique things "the same" ?

If I have an apple and you give me a frog, I have an apple and a frog. They're not the same. If I have apple A and you give me apple B, do I have 2 apples? I have unique apple A and unique apple B. We can pretend that they're the same if you like, but that category is in our thinking, not in the world, and we also know that we can also notice differences between them.

tl;dr the natural world is not fungible, but behaving as if it is, is a convenient abstraction for mathematics and commerce, not a property of the natural objects.

> tl;dr the natural world is not fungible, but behaving as if it is, is a convenient abstraction for mathematics and commerce, not a property of the natural objects.

If it wasn't a property of natural objects (in some way), then how could our predictions work so well in the real world?

What prediction is that? Are you arguing that two rocks or apples or people are actually "can't tell them part" identical?

It works for electrons. But when was the last time that you interacted knowingly with a single electron?

> What prediction is that?

For example, all the predictions that make us capable of building skyscrapers that don't fall down for centuries. Does it matter that two bricks are not "the same piece of matter" if our predictions work the same for both of them? In terms of their behavior under particular circumstances, they are the same.

" In terms of their behavior under particular circumstances" is very specific. "The maths is useful under particular circumstances" is not the same thing as "numbers are real"

Bricks are in the category of "made objects" not natural objects, which generally means that they are _designed_ to come off a production line as similar as humanly possible to the other products. "My iPhone is physically interchangeable to yours" is a statement about the huge efforts of industrial manufacturing to standardise matter, not about the natural world.

Bricks too have quality thresholds that they have to meet or exceed. That alone should tell you that treating them like integers is a convenient abstraction, nothing more. The sibling comment has it right: counting bricks is a great model, but confusing your model for reality is still an old error.

> Does it matter that two bricks are not "the same piece of matter" if our predictions work the same for both of them

No, because of a dense, interconnected web of other social truths (the rest of the arithmetic model), the relative error of this one truth/model is negligible.

However, confusing your model for reality is a fallacy perhaps older than time.

I don't understand how is "the arithmetic model" a social truth, when it clearly corresponds to physical phenomena. You can make a skyscraper that doesn't fall, and it exists regardless of whether other people see it or not. You see it - it's there. What is "social" about that?

> However, confusing your model for reality is a fallacy perhaps older than time.

I think that the human perception the world is a robust enough model to be equated with reality without issues. If you go down the path of denying perception, you might as well go full solipsism, in which case it doesn't even make sense to discuss reality at all.

We have a model of physics which is pretty accurate. The engineers who designed the skyscraper did not even use this model, they used a much simpler one, with known errors. Why? It is simply good enough™. But you can't claim it is even "true" when we know more accurate methods.

My claim is not that the model itself is true - I'm claiming that the underlying mechanisms that rule the world are true and are not subject to change by social consensus.

The model is "good enough" for the purpose of creating a building, but that doesn't make the act of "creating a building" any less real, nor the underlying rules that govern matter any less true. Our descriptions are not real - the rules themselves (which we may not know exactly) are real.

Therefore, mathematics - the set of rules that corresponds to how reality works - itself exists in reality regardless of social consensus. Society can't change them by making a different consensus.

Nothing about mathematics describes how the world works.

As far as rejecting perceptions i think going straight to solipsism is a big jump. We may live in a reality that we have no access to. Donald Hoffman's theories in this area are fun.

Math doesn't do that, models (including language: a model of communication around which we have already reached consensus) do.

2+2=4 stops being true when you use a different method of counting.

That's like saying "I am five foot tall" stops being true when you use a different definition of "tall" (switch it to weight, for example). This isn't insightful and it's also false.

People have agreed on how the plus sign should be defined when it comes to adding two integers, and they've agreed on how to write integers, in which case 2+2 is always four. The only way to change that is to change the meaning of at least one of the four symbols "2", "4", "=", or "+" to some private meaning which no one else shares -- e.g. to distort the language in order to sow confusion, which the use of agreed upon meanings for symbols is designed to eliminate.

Moreover this idea that a sentence changes its truth value the moment someone changes the meaning of one of the words in the sentence is not some deep statement or interesting observation, it's just pointing out that language is useful only if people agree on the meaning of the words.

What's a rock?

What's a "what's"?

Well for either of us to know, we'd need consensus on language, meaning, words, etc.

Nope. The idea of describing an object is built in to humans and likely all mammals.

An inquiry for a description of an object. Nothing in language is difficult it’s finding the smallest abstraction to generate all the rules for all language that is.

You’re asking a simple grammar question under the impression it’s an unsolved science question.

No, I'm merely pointing out the absurdity of asking "what's X" to a needless extent.

Everyone knows what a rock is. Everyone knows what I mean by "throwing two rocks on a pile of two rocks and getting a pile of four rocks" - almost everyone has done it countless times as a child. The "What's a rock" comment is just pointless philosophizing, masquerading as profoundness.

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