The 20th-century mathematician whose name I blanked on in class Wednesday was Kurt Godel. I thought of him during our discussion of "Division by Zero" because his famous achievement, the incompleteness theorem, upset some big applecarts in mathematics. One of Godel's many chroniclers, Rebecca Goldstein, described it to Edge this way:
Most mathematicians, such as David
Hilbert, the towering figure of the previous generation of mathematicians,
and still alive when Gödel was a young man, were formalists. To
say that something is mathematically true is to say that it's provable
in a formal system. Hilbert's Program was to formalize all branches
of mathematics. Hilbert himself had already formalized geometry, contingent
on arithmetic's being formalized. And what Gödel's famous proof
shows is that arithmetic can't be formalized. Any formal system of arithmetic
is either going to be inconsistent or incomplete.
My point was that far from being demoralized by this result, Godel was elated, and for good reason. He was lionized for that work, in life and since. It made his career.
Now that I think about it, though, my initial attempt at a comparison between Godel and the protagonist of "Division by Zero" doesn't go very far. While Godel may have put a stake in the heart of mathematical formalism, he did not subvert the premises of mathematics itself. Instead, he vindicated them, as Goldstein goes on to explain:
Gödel had intended to show that
our knowledge of mathematics exceeds our formal proofs. He hadn't meant
to subvert the notion that we have objective mathematical knowledge
or claim that there is no mathematical proof—quite the contrary.
He believed that we do have access to an independent mathematical reality.
Our formal systems are incomplete because there's more to mathematical
reality than can be contained in any of our formal systems.
Anyone want to argue that the discovery in "Division by Zero," like Godel's discovery, is not a dead end but a newly open door?
I would definitely argue that it gives way to a new horizon.
ReplyDeleteWe saw something similar to this occur in physics with relativity. Newtonian Relativity did not account for objects moving close to the speed of light; therefore, mathematically, it seemed incorrect. It took Einstein's and Lorentz's (plus others) newer concept of special relativity to account for these high velocities and still prove Newtonian mechanics to be correct at semi-normal conditions.
So even though the protagonist manages to prove all numbers equal to each other, it just means our understanding of the universe and mathematics is still incomplete. Everything from basic arithmetic to the upper level calculus would continue work as it is now. Of course, it would be ignorant and naive to know that there was an overarching universal/mathematical theorem that would make sense of the "all-equal numbers problem" and yet choose to not pursue it. The protagonist gave up when in reality she should have been elated at the opportunity to further humanity's understanding of what we know.
If it were possible to prove that 1=2 etc, that would have pretty interesting consequences in terms of mathematical inquiry. The first question that would need to be asked would be, "well why aren't 1 and 2 of a given object the same thing?"
ReplyDeleteI believe like most scientific branches, new mathematic discoveries aren't intended to re-invent the wheel or upset normal life. However, when life altering theorems and such are concluded, most of physics and mathematics just accepts the changes, and shifts accordingly. In daily practice, it would be impossible to prove that 1=2, so everyday life would continue on. The only changes would be relating mathematicians work to the new discovery and somehow integrating the old with the new to continue to describe our world.
ReplyDeletePlease look the papers:
ReplyDeleteThe division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world
Division by Zero z/0 = 0 in Euclidean Spaces
Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh
International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1
-16.
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
http://okmr.yamatoblog.net/division%20by%20zero/announcement%20326-%20the%20divi
http://okmr.yamatoblog.net/
Relations of 0 and infinity
Hiroshi Okumura, Saburou Saitoh and Tsutomu Matsuura：
http://www.e-jikei.org/…/Camera%20ready%20manuscript_JTSS_A…
https://sites.google.com/site/sandrapinelas/icddea-2017
{\bf Abstract: } The common sense on the division by zero with a long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on derivatives we have a great missing since $\tan (\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero. In this book, we will show and give various applications of the division by zero $0/0=1/0=z/0=0$. In particular, we will introduce several fundamental concepts on calculus, Euclidian geometry, analytic geometry, complex analysis and differential equations. We will see new properties on the Laurent expansion, singularity, derivative, extension of solutions of differential equations beyond analytical and isolated singularities, and reduction problems of differential equations. On Euclidean geometry and analytic geometry, we will find new fields by the concept of the division by zero. We will collect many concrete properties in the mathematical sciences from the viewpoint of the division by zero. We will know that the division by zero is our elementary and fundamental mathematics.
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