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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.

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    The last paragraph is too preachy for SE and should be rephrased or deleted. Rigorous truths are dime a dozen, and no one cares about the most. The task of imagination is not to prove theorems but to conceive of deep ones. To that end, "logical tools" are largely irrelevant, and the rigor is useless without the bold conjectures for it to make into truths. Investigations into the heuristics of diagrammatic reasoning are currently booming, both in cognitive science and in philosophy of math, but people long since stopped attributing them to "divinity". – Conifold Aug 28 '19 at 8:00
  • "why do we need geometry?" Because space is fundamental in nature and human life. – Mauro ALLEGRANZA Aug 28 '19 at 8:02
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    What is "pure mathematics" ? Geometry is not mathematics ? What about arithmetic ? we use numbers to count : eggs, money, etc. Why "pure" mathematics must not be useful ? – Mauro ALLEGRANZA Aug 28 '19 at 8:03
  • I apologize @Conifold I did not intend to sound preachy. I am not claiming math should not be useful. But counting and geometric shapes are vastly different conceptions. What is their connection, and is that connection necessary? I believe there may be a connection between shapes and counting, but not a necessary one. It's more of an aesthetic choice because we live in space-time. We can transcend the paradigm and go for pure, divine truth. Using ingredients just as pure might I add. – Justin Latson Aug 28 '19 at 8:36
  • I am uncomfortable with the statement that only geometry could have led Riemann to see far ahead and Weierstrauss is at some sort of disadvantage, because he was more of an algebraist. Can algebraists not see far ahead? Why is geometry the Messiah of God? Why is it the only path? Rebut me, and prove me wrong. I'd love nothing more than to have this passion quelled in a state of absolute truth. I can't explain why this issue gets to me. Perhaps, it is because I see no merit to geometry or perhaps an inferiority complex is more likely. Either way I appreciate your comments! – Justin Latson Aug 28 '19 at 8:41
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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.

  • What is the link between "Roman numerals used for finace" and the OP's question ? – Mauro ALLEGRANZA Aug 28 '19 at 9:58
  • @MauroALLEGRANZA I see it as comparable to the bias towards geometry, even where it's provably identical to algebra – CriglCragl Aug 28 '19 at 12:19
  • Grothendiecks frequently spoke of visions but from reading his notes. You simply see equations and identities. Hmm... Interesting – Justin Latson Aug 28 '19 at 20:11
  • I'd like to add that it is in fact Leibniz whose ideas are closer to the modern calculus than Newton. It is his notation and his approach that we most dearly resemble. Leibniz was basically an algebraist who sought to explicate the rules of a differential ring. Newton was more of a physicist in modern parlance. His fluxion notation was extremely clumsy and full of logical weaknesses. I will concede though that both men arrived at their calculus by considerations of areas of geometric figures and the tangent problem. But was this necessary for Leibniz? – Justin Latson Aug 28 '19 at 21:12
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    My understanding is that Newton did not want his work to be challenged on the basis of the logical defects in calculus, which at the time was brand new and subject to Berkeley's "ghosts of departed quantities" type of criticism. – user4894 Aug 30 '19 at 0:07
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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...

  • I just have to add one thing. Intuition has a history of deceiving us. The shock that mathematicians felt when they learned that not all continuous functions have a derivative is one example I'd cite. Although Poincare claimed that the source of the continuum is geometry. And therefore all math is essentially geometric seeing as to how infinity is essential to math. O still feel as if this is not necessarily true. Name one problem that algebra can't solve but geometry can. Or, heck, name a concept. I can name you multiple that geometry can't. So can history. – Justin Latson Aug 28 '19 at 20:09
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    Notoriously, finding the root of an arbitrary fifth dimensional polynomial is possible only if you bring in real analysis or trigonometry -- both are based in geometry, algebra does not, in itself, allow for arbitrarily close approximations and the fifth degree does not have a closed form, because of Abel's Theorem. – user9166 Aug 28 '19 at 20:39
  • And (maybe unique among my posts on math) I did not mention intuition. I mentioned derivation and exposition of results. Employing more means and a greater range of sensory models facilitates derivation, and any kind of exposition that we can find to be well-grounded is useful. No one of them is really better than any other. Detecting when an algebraic proof is wrong is really not easier than detecting when one assisted by geometry is wrong. – user9166 Aug 28 '19 at 20:47
  • And that's exactly my point. Approximation is not a solution. It is a mere shadow in Plato's cave to the real forms behind them. We can only approximate better and better with less of a margin of error the more times we iterate a certain computational identity. We will, however, never attain it. I also disagree that real analysis is not based on algebra. How would a power series look without the operations or the integers? Trigonometric identities can be formulated by algebraic means as well. Hey are simply one in an infinite number of combinatorical logic. – Justin Latson Aug 28 '19 at 21:05
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    This has gone on too far. I think your whole point here is bigotry against what does not appeal to you, and you are not open to that. But the fact one can or might scrape the geometry off calculus does not mean one should just so folks who prefer computation feel safer. Who cares whether we have seven different ways of looking at things, as long as they help? Well, only folks who would rather have control that discoveries and who disapprove of others' methods: In other words impractical bigots. – user9166 Aug 28 '19 at 22:17
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  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.

  1. The point is effectively just "space". You cannot seperate counting from this basic assumption.
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    Excellent answer! You've provided a legitimate grounds for the necessity of geometry in maths. I will think on this further. Thank you. – Justin Latson Aug 29 '19 at 1:04
  • Thanks justin, I piss alot of people off here...if you can plus 1 it so I can get a rep of 125 points and start down voting bad questions and advice...I would appreciate it. Joking aside (it's not really a joke), I can expand the answer if you wish. – Eodnhoj7 Aug 29 '19 at 1:32
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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.

  • Why do we need geometry then? – Justin Latson Aug 28 '19 at 8:32
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    @JustinLatson Because it's easy to reason about a circle and how a line can possibly intersect it, but harder to reason about two equations and where they happen to have the same result. Geometry can help one reason differently, in the same way pure algebra can help one reason differently. The fact they're provably interchangeable means it doesn't matter in the end which you use to come to a conclusion, so the flexibility in how you can think about the problem (or solution) is a useful tool. – Delioth Aug 28 '19 at 17:53

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