Why do we need geometry for pure math?

Question

Karl Weierstrass had a very interesting critique of Riemann's work. Supporters of Riemann, claim that a pure logician would never have been able to see the things that the "geometric imagination" of Riemann's could. While the Weierstrassians valued "algebraic truth".

In the current state of math, I think most working mathematicians would agree that Riemann probably saw further ahead than Weierstrass and for the majority of his life did not have the logical tools to prove what he knew was right. This was a task for those after him. Unlike Weierstrass who mostly just continued on with the existing math of the time.

However, this claim that the "geometric imagination" is what allowed him to see further and that it is of a divine character is somewhat flaky. I see no reason why Weierstrass couldn't have done the things Riemann did you, and what more with the added benefit of being rigorous. So then, I ask all the philosophers and pure mathematicians, why do we need geometry?

Have pure mathematicians been wasting their time on something which provides no rigorous truth when they should be investing in algebraic truths? Have we been deceiving ourselves with the elegance of geometry into thinking this signifies some sort of divinity? Does this mean a blind man could only be a logician and never see ahead into math beyond his time?

P.S. for the applied mathematicians this question is not for you. Geometry is plenty useful for the physical world.

Answer 1

Newton, with the bias of his time, sought to formulate all his proofs in geometrical terms. He was also a freemason, a cult-tradition that identifies divinity with architecture, and geometry - and that tradition of divine geometry goes back at least to Pythagoras. Newton had to do a great deal of work to recast things in that way. Was it a waste of time? It wasn't just about ideas of divinity, he had to follow his intuition, and it got him largely where he wanted to go (minus the alchemy).

Finances were conducted in Roman numerals for hundreds of years after Indo-Arab ones were used for everything else, because they were felt to somehow be more reliable. Imaginary numbers still face opposition and problematising because they contradict our intuition, even though the generalisation of octonions is increasingly being looked to in explaining the exact array of subatomic particles we have. And mathematics of non-continuous, quantised, materials is still looked askance at, as somehow darkly sorcerous.

Symmetry has proven to be probably the most powerful insight in physics. Elegance, and especially beauty, are often derided as heuristics, but frequently cited by deep thinkers and seen as manifested in their work.

It's easy to think, why privelege geometry. But why privelege any type of mathematics, and put the insights and proofs from other branches on a lower pedastal? They are tools to think with, it's not about reifying the tools, but the thinking.

It's worth pointing out around a third of our brain's total power is dedicated to visual processing. I wonder what kind of proofs dolphins will generate, once we share mathematics with them - I suspect they will have deeper intuitions than us about the behaviour of fields.

Answer 2

Why do we need algebra? Are you writing off Greek math as not math yet? Surely being able to leverage another sense for analogies is going to be a useful adjunct to insight. (And that sense is more kinesthetics than sight, there are good blind geometers.)

The mathematicians of the Cartesian era added algebra to geometry, not the other way around. It gave language a firmer grip upon the field, so that we could work with direct computation, language and our sense of space. Why are you not asking why they bothered to do that?

This whole notion that the linguistic part of math is the most certain was pursued to its logical conclusion by the logicians. We got Goedel. Ultimately, it pointed out its own weaknesses. And the claims it has to ultimate clarity and completeness are illusions.

But we are stuck with the bias that caused that all to begin with. And we should shake it. Those same logicians later established "nonstandard methods", vindicating infinitesimals as a useful method of reasoning, and disambiguating the problem that created Wierstrass's entire enterprise.

Language, being originally designed for the transfer of information makes algebraic math easier to feel certain of when it is shared. But for establishing a result we often draw pictures, move our bodies, imagine distortions of a visual field or of a piece of matter. We make a topology out of a family of logics. We reinforce our statements about a categorical map with graphs. We take quadratic constraints and we talk about hyperbolic paraboloids. The leverage on the sense of sight and touch helps with the facility of sharing and the accessibility of our result. All levers are levers, all ways of grasping a concept tighter have value.

An entirely geometric explanation may be breathtakingly concise to some but not "feel good" to those who do not immediately communicate in this mode well. Who cares? It is no less rigorous than any other kind of mathematical explanation, due to its form. All mathematics is ultimately exposition of wholly internal insights. It is all analogy. We are free to discuss things in terms of territorial kittens, if it makes the point...

Answer 3

1. Numbers are means for quantifying reality.

2. This quantification requires counting, where one phenomenon is separated from another or united: In counting 2 oranges we unite them into one set in one respect while observing that 1 orange exists in a variety of states as 2 oranges in a seperate respect.

3. Numbers do not exist without counting, and the most basic form of counting is observing symbols.

The most basic symbol is the point considering all empirical phenomena exist as a point in the distance or are composed of points up close (ie jagged edges in the curves). Also it is a universal axiom that represents space and we use intuitively to measure time.

4. The point is effectively just "space". You cannot seperate counting from this basic assumption.

Answer 4

for every conception of geometry there is an associated system of algebra which captures the mathematical truths contained in that geometry. as such, the two fields are "joined at the hip", and mathematicians can freely switch back and forth between the conception of geometry as lines and points in space, etc. and the conception of (for example) systems of linear equations in three independent dimensions.

I cannot second-guess Riemann nor Weierstrauss, and leave those issues to those trained in the history of the field.