Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences.
>On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs.
Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition.
The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.
The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).
I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.
In the end arguing about whether mathematics is a science or not makes no more sense than bickering about tomates being fruit; can be answered both yes and no using reasonable definitions.
[1] And even this has limits: https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...
But we can be more sure of the deductive validity of a proof than we can be of any of the claims you make in these sentences, so I don't think they can serve to establish any doubt. If we're wrong about deductive logic, then we can only be more wrong about any empirical claims, which rely on deductive logic plus empirical observations
When we try to model something probabilistically, it is usually not a great idea to model the probability that we made an error in our probability calculations as part of our calculations of the probability.
Ultimately, we must act. It does no good to suppose that “perhaps all of our beliefs are incoherent and we are utterly incapable of reason”.
In practice when proofs of research mathematics are checked, they go out to like 4 grad students. This isn't a very glamorous job for those grad students. If they agree then it's considered correct...
But note this is just the bleeding edge stuff. The basic stuff is checked and reproven by every math undergrad that learns math. Literally millions of people have checked all the proofs. As long as something is taught in university somewhere, all the people who are learning it (well, all the ones who do it well) are proving / checking the theory.
Anyway, when the scientific community accepts a bad proof what effectively happens is that we've just added an extra axiom.
Like when you deliberately add new axioms, there are 3 cases
- Axiom is redundant: it can be proven from the other axioms. (this is ... relatively fine? we tricked ourselves into believing something that is true is true, the reason is just bad.)
This can get discovered when people try to adapt the bad proof to prove other things and fail.
Also people find and publish and "more interesting", "different" proofs for old theorems all the time. Now you have redundancy.
- Axiom contradicts other axioms: We can now prove p and not p.
I wonder if this has ever happened? I.e. people proving contradictions, leading them to discover that a generally accepted theorem's proof is incorrect. It must have happened a few times in history, no?
o/c maybe the reason this hasn't happened is that the whole logical foundation of mathematics is new, dating back to the hilbert program (1920s).
There are well known instances of "proofs" being overturned before that, but they're not strictly logically proofs in the hilbert-program sense, just arguments. (Of course they contain most of the work and ideas that would go into a correct proof, and if you understand them you can do a modern proof)
e.g. https://mathoverflow.net/a/35558
Cauchys proof that, if a sequence of continuous functions converges [pointwise] to a function, the limit function is also continuous (cauchys proof only holds for uniform convergence, not pointwise convergence - but people didnt really know the difference at the time)
- Axiom is independent of other axioms: You can't prove or disprove the theorem.
English doesn't have a "I'm just hypothesizing all of this" voice, if it did exist this post should be in it. I didn't do enough research to answer your question. Some of the above may be wrong, e.g. the part about the 4 grad students. One should probably look for historical examples.
Math is scientific in the sense that you've proposed a hypothesis, and others can test it.
Also the empirical part means natural phenomena needs to be involved. Math can be purely abstract.
The incompleteness theorem doesn't say that there are statements which are unprovable in any absolute sense. What it says is that given a formal system, there will always be statements which that particular formal system can't prove. But in fact as part of the proof, Godel proves this statement, just not by deriving it in the formal system in question (obviously, since that's what he's proving is impossible).
The way this is done is by using a "metalanguage" to talk about the formal theory in question. In this case it's a kind of ambient set theory. Of course, the proof also implies that if this ambient metalanguage is formalized then there will be sentences which it can't prove either, but these in general will be different sentences for each formalized theory.
The "symbol pushing" is a methodological tool, and a very useful one that opened up the possibility of new expansive fields of mathematics.
(Of course, it is important to always distinguish between properties of the abstraction or the tool from the object of study.)
> Euclid's Elements is 2300 years old and is presented in a completely abstract way.
depends on what you mean by completely abstract. Euclid relies in a logically essential way on the diagrams. Even the first theorem doesn't follow from the postulates as explicitly stated, but relies on the diagram for us to conclude that two circles sharing a radius intersect.
This is a thought-provoking paper on the issue by Viktor Blasjo, Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry https://link.springer.com/article/10.1007/s10699-021-09791-4
which was recently the subject of a guest video on 3blue1brown https://www.youtube.com/watch?v=M-MgQC6z3VU
https://math.stackexchange.com/questions/31859/what-concept-...
Other great sources for quick intuition checks are Wikipedia and now LLMs, but mainly through putting in the work to discover the nuances that exist or learning related topics to develop that wider context for yourself.
I may be off-base as an outsider to mathematics, but Euclid’s Elements, per my understanding, is very much grounded in the physical reality of the shapes and relationships he describes, if you were to physically construct them.
I am going to quote from the _very beginning_ of the elements:
Definition 1. A point is that which has no part. Definition 2. A line is breadthless length.
Both of these two definitions are impossible to construct physically right off the bat.
All of the physically realized constructions of shapes were considered to basically be shadows of an idealized form of them.
The complex number system started being explored by the greeks long before any notion of the value of complex spaces existed, and could be mapped to something in reality.
I think dominating on a first date is a risk (which I was mindful of) but just being yourself, and talking about something you're truly passionate about is the key.
I tried using if for this: https://adventofcode.com/2023/day/12 but computer said no
Turns out I’m neither good in maths nor teaching
Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.
That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellectual leaps” which are encoded into axioms, and everything in math which is provable is only provable based on a particular chosen set of axioms.
It may be true that math beyond that basic level doesn’t make a point of going back and explicitly reviewing that point, but it is just kind of implicit in everything later.
Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
QED
If she laughs at that kind of thing, I can see why you married her.
https://plato.stanford.edu/entries/mathematics-constructive/ is one place that you could start filling in that gap.
For sure there are valid arguments on whether or not to use certain axioms which allow or disallow some set theoretical constructions, but given ZFC, is there anything that follows that is unprovable?
In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with weirder things. Such as well-specified finite problems that provably have a polynomial time algorithm to solve...but for which it is impossible to find or verify that algorithm, or put an upper bound on the constants in the algorithm. In what sense does that algorithm exist, and is finite?
Does that sound impossible? An example of an open problem whose algorithm may have those characteristics is an algorithm to decide which graphs can be drawn on a torus without any self-crossings.
If our notion of "exists" is "constructable", all possible mathematical things can fit inside of a countable universe. No set can have more than that.
Also this: https://arxiv.org/pdf/1212.6543
Assuming you haven't looked at these already, of course.
Numbers, for example, are abstract in the sense that you cannot find concrete numbers walking around or falling off trees or whatever. They're quantities abstracted from concrete particulars.
What the author is concerned with is how mathematics became so abstract.
You have abstractions that bear no apparent relation to concrete reality, at least not according to any direct correspondence. You have degrees of abstraction that generalize various fields of mathematics in a way that are increasingly far removed from concrete reality.
Leibniz (late 1600s) helped to popularize negative numbers. At the time most mathematicians thought they were "absurd" and "fictitious".
No, not highly abstract from the beginning.
Wasn't that imaginary numbers?
https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theor...
Geometry is “attached” to the physical world… but in an abstract way… but you can point to the thing your measuring maybe so it doesn’t count…
Abstraction was perfected if not invented by mathematics.
It wasn't; but that's a common misunderstanding from hundreds of centuries of common practice.
So, how has maths gotten so abstract? Easy, it has been taken over by abstraction astronauts(1), which have existed throghout all eras (and not just for software engineering).
Mathematics was created by unofficial engineers as a way to better accomplish useful activities (guessing the best time of year to start migrating, and later harvesting; counting what portion of harvest should be collected to fill the granaries for the whole winter; building temples for the Pharaoh that wouldn't collapse...)
But then, it was adopted by thinkers that enjoyed the activity by itself and started exploring it by sheer joy; math stopped representing "something that needed doing in an efficient way", and was considered "something to think about to the last consecuences".
Then it was merged into philosophy, with considerations about perfect regular solids, or things like the (misunderstood) metaphor of shadows in Plato's cave (which people interpreted as being about duality of the essences, when it was merely an allegory on clarity of thinking and explanation). Going from an intuitive physical reality such as natural numbers ("we have two cows", or "two fingers") to the current understanding of numbers as an abstract entity ("the universe has the essence of number 'two' floating beyond the orbit of Uranus"(2)) was a consequence of that historical process, when layers upon layers of abstraction took thinkers further and further away from the practical origins of math.
[1] https://www.joelonsoftware.com/2001/04/21/dont-let-architect...
That is, numbers were specifically used to abstract over how other things behave using simple and strict rules. No?
Agree that math is built on language. But math is not any specific set of abstractions; time and again mathematicians have found out that if you change the definitions and axioms, you achieve a quite different set of abstractions (different numbers, geometries, infinity sets...). Does it mean that the previous math ceases to exist when you find a contradiction on it? No, it's just that you start talking about new objects, because you have gained new knowledge.
The math is not in the specific objects you find, it's in the process to find them. Rationalism consider on thinking one step at a time with rigor. Math is the language by which you explain rational thought in a very precise, unambiguous way. You can express many different thoughts, even inconsistent ones, with the same precise language of mathematics.
The tendency towards excessive abstraction is the same as the use of jargon in other fields: it just serves to gatekeep everything. The history of mathematics (and science) is actually full of amateurs, priests and bored aristocrats that happened to help make progress, often in their spare time.
To put it another way: Jargon is the source code of the sciences. To an outsider, looking in on software development, they see the somewhat impenetrable wall of parentheses and semicolons and go "Ah, that's why programming is hard: you have to understand code". And I hope everyone here can understand that that's an uninformed thing to say. Syntax is the easy part of programming, it was made specifically to make expressing the rigorous problem solving easier. Jargon is the same way: it exists to make expressing very specific things that only people in this subfield actually think about easier, instead of having to vaguely gesture at the concept, or completely redefine it every time anybody wants to communicate within the field.
People are aware that you need context to motivate abstractions. That's why we start with numbers and fractions and not ideals and localizations.
Jargon in any field is to communicate quickly with precision. Again the point is not to gatekeep. It's that e.g. doctors spend a lot of time talking to other doctors about complex medical topics, and need a high bandwidth way to discuss things that may require a lot of nuance. The gatekeeping is not about knowing the words; it's knowing all of the information that the words are condensing.
Formal reasoning is the point, which is not by itself abstraction.
Someone else in this discussion is saying Euclid's Elements is abstract, which is near complete nonsense. If that is abstract our perception of everything except for the fundamental [whatever] we are formed of is an abstraction.
What do you think "formal" means in that sentence.
It means "formal" from the word "form". It is reasoning through pure manipulation of symbols, with no relation to the external world required.
Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far it moved... I am missing something?
To resolve the paradox, you have to show what's wrong with the reasoning, not just observe the obviously false conclusion.
Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos - and came up to something that wasn't considered a function - a square wave.
The focus on abstraction and axiomatization came after decades of trying to repair mathematics over and over again. Trying to retell the story in terms of the resulting mathematical flow of the ideas, completely mangles the actual flow of events.
> forced there by a series of crises where the mathematics that they knew fell apart
This can be said to be true of those working in foundations, but the vast majority of mathematicians are completely uninterested in that! In fact, most mathematicians today probably can't cite you the set-theoretic (or any other foundation) axioms that they use every day, if you ask them point-blank.
Math in its core has always been abstract. It’s the whole point.
I don't think so. E.g. there may be some abstractions in numerical linear algebra, but the subject matter has always been quite concrete.
What you call concrete - were the origins of math as we know it. Geometry, astronomy, metaphysics etc they all had in common fundamental abstract thing that we call math today.
Saying “math got abstract” is like saying “a tree got wooden”. Because when it was a seed - it wasn’t yet a tree in a full sense.
Given the collective time put into it, easier stuff was already solved thousands of years ago, and people are not really left with something trivial to work on. Hence focusing on more and more abstract things as those are the only things left to do something novel.
But also wrong, the easier stuff was solved INCORRECTLY thousands of years ago. But it takes advanced math to understand what was incorrect about it.
- they are material objects
- they are concepts I understand
- they are sequences of letters
- they are English words
- ...
Not sure why oneness is privileged as what they have in common, and their oneness is meaningless by itself. Oneness is a property that is only meaningful in relation to other concepts of objects.
However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers or elements, and instead focus on the relationships and operations themselves. For me, this reveals an even deeper beauty, i.e., different domains of mathematics, or even problems in computer science, can be unified under the same algebraic framework.
Consider, for example, the fact that the set of real numbers forms a vector space over the set of rationals. Can it get more abstract than that? We know such a vector space must have a basis, but what would that basis even look like? The existence of such a basis (Hamel basis) is guaranteed by the axioms and proofs, yet it defies explicit description. That, to me, is the most intriguing kind of abstractness!
Despite being so abstract, the same algebraic structures find concrete applications in computing, for example, in the form of coding theory. Concepts such as polynomial rings and cosets of subspaces over finite fields play an important role in error-correcting codes, without which modern data transmission and storage would not exist in their current form.
Also, I don’t think ZF sans the axiom of infinity works as an ultrafinitistic theory? It still has every natural number, just not the set of all of them.
We used Peano arithmetic when doing C++ template metaprogramming anytime a for loop from 0..n was needed. It was fun and games as long as you didn't make a mistake because the compiler errors would be gnarly. The Haskell people still do stuff like this, and I wouldn't be surprised if someone were doing it in Scala's type system as well.
Also, the PLT people are using lattices and categories to formalize their work.
"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies."
-- Stefan Banach
More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.
I get what they're saying in practice. But numbers are abstract. They only seem concrete because you'd internalized the abstract concept.
We likely need new mathematics for making progress in physics or ..say.. have a better understanding of the PvsNP kind of problems, but very few high caliber mathematicians are motivated to do this.
Which makes sense, as it’s way easier and prestigious to define and solve your own abstract problems, publish one paper per grad student per year and coast through research life.
On the other hand, two cookies plus three cookies, what even is a cookie? What if they're different sizes? Do sandwich cookies count as one or two? If you cut one in half, does you count it as two cookies now? All very abstract. Just give me some concrete definitions and rules and I'll give you a concrete answer.
The Peano axioms are pretty nifty though. To get a better appreciation of the difficulty of formally constructing the integers as we know them, I recommend trying the Numbers Game in Lean found here: https://adam.math.hhu.de/
When I was studying, I always got top marks in Analysis.
Then came Algebra, Topology and similar nightmares. Oh crap, that was difficult. Not really because of the complexity, but rather because of abstraction, an abstraction I could not take to physics (I was not a very good physicist either). This is the moment I realized that I will never be "good in maths" and that will remain a toolbox to me.
Fast forward 30 years, my son has differentials in high school (France, math was one of his "majors").
He comes to me to ask what the fuck it is (we have a unhealthy fascination for maths in France, and teach them the same was as in 1950). It is only when we went from physical models to differentials that it became clear. We did again the trip Newton did - physics rocks :)
I personally cannot wrap my head around Cantor's infinitary ideas, but I'm sure it makes perfect sense to people with better mathematical intuition than me.
