r/mathematics • u/Recent-Day3062 • 1d ago
A question about how “algebra” keeps expanding and the like
Be kind. I’m a physicist who has more interest in abstract math than most colleagues.
Obviously, when basic algebra was invented, it makes perfect intuitive sense. We can all see that if two numbers are equal and you add the same amount those numbers are equal. Same for commutative. The distributive property is more working the algebra.
So it’s surprising to me it had anywhere “to go”. For example, why should matrix algebra work? And other algebraic structures.
We think of math starting with concrete numbers, then go to matrices and tensors. And loads of other examples.
But why should the idea be able to expand?
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u/0x14f 1d ago
We started to study a larger collection of objects, structures and spaces which have an algebraic description. That's what the expansion is in your analogy.
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u/Recent-Day3062 1d ago
Well let me put it this way.
Obviously the larger space always existed. So did we just see a very simple example and that gave us clues as to what more might exist? I guess this is almost a history question…
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u/wollywoo1 1d ago
I couldn't give you the historical rundown very accurately. But, we can imagine. If you rotate a point by 90 degrees about the origin, and then reflect it horizontally, you don't ge tthe same thing as when you reflect horizontally and then rotate it by 90 degrees. Well, that's a clue that there are operations that aren't commutative, and if you are clever enough that will lead you into the rabbit hole of matrix algebra, the symmetric group and so on.
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u/AcellOfllSpades 1d ago
Obviously the larger space always existed
"Obviously" is a very loaded word here. There's endless debate over whether math is invented or discovered.
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u/Intrepid-Struggle964 18h ago
Discovered.. it cant be invented when most of math comes out with a single tick of n+1. So unless it was invented by the sense of taking n an adding ressesion to it imma go with math was here before any human ever thought to use it.
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u/lemniscateall 1d ago
A lot of the time a new structure is noticed, one big question tends to be "how much does this resemble R (or R^n)?" Say you're in the practice of solving systems of linear equations, over and over, and you notice that all you're really doing is manipulating the coefficients of the variables; then you come up with a shorthand way of expressing those relationships. And you start to mess with that shorthand notation, and you ask, can I add these, scale them by constants, and generally do the things I can with vectors?" And so forth.
Rather than looking at basic algebra and thinking of extending it, we generally find something and figure out how it resembles something we already know about. Often, it's the imperfect resemblance to a known structure that makes a new one interesting; if it's totally like or totally unlike known structures, that tends to limit both its utility and the interest it holds for mathematicians. So, matrices have nice addition and scalar multiplication that resemble our basic numbers, but then the parallel fails at the commutativity of matrix multiplication.
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u/AcellOfllSpades 1d ago
Why shouldn't it?
We build structures that intentionally preserve the laws we like.
Like, ℝ² as a vector space has commutative and associative addition because ℝ does. ℝ² "inherits" that property from ℝ.
And we build structures that preserve some of those laws; when the laws aren't preserved, like commutativity in matrix multiplication, then we have to make some changes to how we do things. But many basic laws have some analogue when talking about any type of "mathematical object", and so we can do something pretty similar to the algebra that we're familiar with.
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u/psychophysicist 1d ago
Once you have the insight that algebra "works" because of distributive and commutative properties. you can turn that insight around. if you can come up with some objects and operators on them that satisfy the distributive and commutative properties, then you can do algebra on those objects.
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u/Recent-Day3062 1d ago
Yeah, but what leads you to find them? I mean it’s a wonderful coincidence that there are extensions, they work, and we found them.
I just finished Linear Algebra done Right and still to absorb it all. I must say from that that his basic scheme is to assert there are vectors, that live in vector spaces, and linear transforms - and it all comes from a basic definition of a vector then the long-hand basic algebra.
But the motivation there may have been from basic multivariate calculus for solving practical problems
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u/ascrapedMarchsky 1d ago
Many times already men have thought that they had solved all the problems, or at least that they had made an inventory of all that admit of solution. And then the meaning of the word solution has been extended; the insoluble problems have become the most interesting of all, and other problems hitherto undreamed of have presented themselves. For the Greeks a good solution was one that employed only rule and compass; later it became one obtained by the extraction of radicals, then one in which algebraical functions and radicals alone figured. Thus the pessimists found themselves continually passed over, continually forced to retreat, so that at present I verily believe there are none left.
- The Future of Mathematics, Henri Poincaré
The lack of a general quintic formula led to Galois theory. In its nascent form the theory was limited to finite field extensions. The problem with infinite extensions was that all subgroups of the Galois group no longer necessarily arose as subgroups fixing subextensions. Richard Dedekind keenly observed, however, that the Galois group was something like a manifold. This insight eventually allowed the theory to encompass infinite extensions and led to Grothendieck's topological reformulation of Galois theory in terms of étale algebras. A new problem that emerged from this theory was that topological spaces were poorly behaved categorically. In the last decade algebraic geometers have thus started to replace topological spaces with condensed sets.
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u/Intrepid-Struggle964 18h ago
Idk from my finding everything seems to already be structured from n, now just keep adding 1 an see how much all of math falls out of it without adding anything more. The problem is assuming we made the math work. When reality is just math unfolding constantly. Algebra is a way to take succession an use it on ur own terms.
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u/piranhadream 1d ago
The basic answer to your question is just that the desirable, intuitive properties of the integers don't suffice to uniquely identify the integers as a number system. For instance, the integers modulo 2 (or modulo n) also satisfy the same properties, and this naturally leads one to ask how you can distinguish between these two number systems, and what properties do they gain or add as a consequence of this distinction?
For instance, in the integers, if ab=0, then a=0 or b=0. But in the integers modulo 6, this is no longer true: 2*3 = 6, which is congruent to 0 (mod 6). 2 and 3 are what we call zero divisors, and it's not hard to convince yourself that the integers mod n (n > 1) has zero divisors if and only if n is composite.
So part of what happens in algebra is that we try to classify these objects, called rings, with early classifications essentially trying to answer the question, "How many properties does this ring share with the integers?" This gives rise to all sorts of rings -- commutative rings, fields, division algebras, integral domains, Euclidean domains, principal ideal domains, unique factorization domains, Ore domains, etc. In a lot of cases, the "niceness" of a given ring correlates to nice properties of objects associated with the ring, giving us a way to study a different, possibly more difficult object by instead looking at a ring.
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u/Recent-Day3062 1d ago
Yeah, but why in gods name did anyone start doing modulo arithmetic ?
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u/piranhadream 1d ago
Things with discrete periodic behavior, like clocks and calendars, have been around for centuries and are naturally modeled with modular arithmetic.
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u/Recent-Day3062 19h ago
Yes, but doing multiplication and division makes no sense. And as to addition and subtraction, no one would be interested in the abstraction. All anyone wants to know is the 10 hours after 10pm is 8am
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u/Intrepid-Struggle964 18h ago
Multiplication an division fall out of succession addition subtraction as well. I think the thing is why did anyone advance number types. Not so much why does it work the way it does
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u/Intrepid-Struggle964 18h ago
They didnt it legit comes from peono math its already there. Its finding structure in numbers an naming it
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u/dld2517 1d ago
It expands because of abstraction, number theory, object notation and the axiomatic nature of each of these. If I use abstraction to generalize the properties of a given numeric object, the axioms do not cease to be true outside the specification, that is we still have definitions in the abstracted space. So the ideas of commutation, associativeness, distributiveness. Orthogonality, and ordinality exist and extend into each abstraction layer above.
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u/Intrepid-Struggle964 18h ago
Id disagree axioms are human made .. nor are all of them needed
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u/dld2517 16h ago
Axioms are truths. If they are disproven or untrue they are not axioms. A person can discover them but not create them.
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u/Intrepid-Struggle964 16h ago
Axioms aren't claims that are true or false on their own. They're starting assumptions, and whether they're "true" depends on what you're modeling. Take the parallel postulate: it holds on a flat plane and fails on a sphere. Neither geometry is wrong, they're just different systems. Same with the axiom of choice: ZF works with it or without it, and mathematicians study both. Nobody "disproved" the old axiom, we just found that the system still works when you swap it out.
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u/TUVegeto137 23h ago
If you're a physicist, then how can you not understand the need for matrices? 😂
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u/Recent-Day3062 18h ago
That’s my point. I understand. I’m just trying to figure out how someone found N algebraic think like this
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u/Intrepid-Struggle964 16h ago
Algebra expands because it's really the study of rules, not numbers. Anything that obeys the same rules inherits the same results. Matrices obey most of the number rules (but not commutativity), which is why the algebra mostly carries over. And a lot of arithmetic, including addition, multiplication, and mod, can be built from just "next number" repeated or wrapped. Once you have a system, the natural question is how you can manipulate it, and that question is algebra.
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u/schungx 15h ago
People started discovering that the normal algebra rules are not restricted to numbers. They carry over to other things, and a LOT of things in nature reflect the same rule.
So now you get algebras of a lot of stuff, most of them have nothing to do with numbers, but they BEHAVE like numbers.
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u/sighthoundman 1d ago
I like to think of algebra as the study of symbol manipulation. So we keep pushing the boundaries to see if manipulating symbols in one particular way can describe more things, or if we can do it in "weirder" circumstances, and so on.
The ancient Greeks, and the Arabs after them, did their mathematics verbally. They didn't use the quadratic formula, they completed the square. In fact, one of the historically important treatises on mathematics was al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah (usually referred to as al-Jabr, the source of our algebra) by al-Khwarizmi, Latinized as Algoritmi and the source of our word algorithm. That long Arabic phrase means The Concise Book of Calculation by Restoring and Rebalancing, which mostly referred to moving terms from one side of the equation to the other. (Much more difficult when you don't have an equation written out in symbols.)
Over time we started doing things more and more symbolically, and developed the quadratic formula for solving quadratic equations. Then Tartaglia and del Ferro figured out how to solve the cubic, and Cardan used that work to figure out how to solve the quartic. But there's a problem with the cubic: we're taking cube roots of negative numbers and adding them and we get normal numbers as a result. Bombelli came up with a crazy idea: So what? Just treat them like numbers, and (if you're careful) everything works out.
From there, Lagrange and Abel and Ruffini were messing around with permuting the roots of polynomials to prove that there isn't a formula for solving the quintic that just uses the 4 basic numerical operations and taking n-th roots. Galois extended this a lot and all of a sudden we have group theory. The Hamiltonian group came about because Hamilton was trying to extend complex numbers (where multiplication by i corresponds to a 90-degree rotation) to 3D. It eventually lost out to vector notation (for most applications).
But we're moving things around in 3-space (or, when we're doing fluid mechanics, 6-space, just because), so we needed a convenient notation for maps from n-space to n-space, and a convenient way to talk about n-space. So vector spaces and linear transformations just kind of arose organically. And matrices just happen to be a convenient way to represent linear transformations on finite dimensional vector spaces.
TL;DR: You can go on a very long journey one step at a time. And because we normally teach it without any historical context, it can feel like "why would anyone do this?" But each step makes sense to someone, based on where they are at the time. We teach the ones that turn out to be useful somewhat often, and the others hide in journals until they're forgotten. (See prosthaphaeresis. It was an important calculation aid for about 25 years until logarithms were invented. No one knows about it today.)