Do the numbers, Einstein: AI is more than maths as some know it
Why logic is driving graph databases
Microsoft arrived on the graph-database scene last month. Three years in the making, Microsoft released Trinity under a typical-by-now-of-Microsoft-boring-trade name of Graph Engine.
Already on that scene are Neo4J, MarkLogic, Oracle, SAP and Teradata - among others.
Driving Microsoft, like those before, is the desire to connect - to establish connections between things and derive some kind of gain.
Those “things” could be people, “likes”, online sales – tech firms are almost literally trying connecting the dots or as they like them to be called “nodes.”
But that’s so last year. The new thing is Artificial Intelligence and the Machine Learning that gets us there. And, slowly, tech firms are pushing graph engines as an enabler of ML.
Neo4J, for example, offers Graphify an unmanaged extension for language text classification. Microsoft’s Concept Graph, here, is build on Probase that uses Trinity for natural language and machine-language understanding.
All that ML and Artificial Intelligence (AI) jazz is directly founded on mathematics and while you’ll no doubt have heard much hype about ML and AI and talk of tools and frameworks, you are missing the point – or, rather, getting a little ahead of the game.
In fact, nothing demonstrates the need to familiarise your self with mathematics better than graph databases and nothing demonstrates that better than the City of Kaliningrad on the Baltic in Russia.
But before we get there, the first thing you have to realise is that “maths” is not “arithmetic” as many might believe. I was never taught this at school (or I was away that day) so I confused the two for years. Now I know that arithmetic is doing sums and maths is about understanding and using the patterns that occur naturally in numbers. It can and often does involve equations but they are simply a shorthand way of expressing ideas. So, maths is about using the patterns and maths research is about finding those patterns in the first place. And some of that research has revolutionised our lives.
Some maths research starts with questions that appear to have no connection to numbers at all. For example, consider the city of Kaliningrad. A river runs through it and forms two islands and the city is built both on the banks of the river and on the two islands.
Now, back to Kaliningrad, or rather, Königsberg as it was named in the 18th century, which is where we must travel. The city is cut in half by a river, the Pregoyla, and in the Pregoyla are two islands. Some in the city had become fascinated by what would later became maths-based question: was it possible to cross each of the seven bridges in the city just once in one journey? Were talking four land masses with four bridges connecting one island to the mainland - two North and South), two bridges connecting the second island (one North and one South), and one bridge connecting the two islands – East and West.
You can of course try to draw a path and fail.
But failing doesn’t prove that it can’t be done. Succeeding proves that it can be done but no one could succeed. Swiss mathematician, physicist, astronomer, logician and engineer Leonhard Euler became interested in the problem and he did what mathematicians do: he reduced an apparently complex problem to a set of basic components and proceeded to derive the rules by which they operated.
This approach meant that he (and now we) could not only solve the Königsberg problem but solve it for any number of bridges in any city. Better still this led to the development of what we now call graph theory and graph databases - which are so generalised that they can be used for space research.
Euler realised that the shape of the land was immaterial to the problem, so he reduced the land masses to a set of points or what we would now call (in a graph database) nodes. The bridges are simply links between the nodes and are now called edges.
Once simplified, Euler reasoned as follows. If one node has two edges and you start your journey on the node then you have to cross one edge to get off it (1).
You can then wander around as much as you like, visiting other nodes and crossing other edges but ultimately you have to cross the other edge (2) and you are now stuck on the node with no way to get off.
To put that more simply, if you start on a node with two edges you have to finish on it. (To translate back to the Königsberg bridge problem, if you start on one land mass with two bridges, you have to finish on that land mass.) This turns out to be true for all nodes with an even number of edges. The sequence: off, on, off, on, off, on always has to end with on.
And by the same token, if you don’t start on a node with two edges you can cross onto it but you have to cross the other edge on your way off, so if you start off on a node that has two edges, you have to also finish off the node. (In Königsberg speak, if you start off on a land mass with only two bridges, you have to finish off that land mass.) And this is also true for all nodes with an even number of edges.
And, as you have already guessed, the rules are reversed if a node has an odd number of edges. If you start on a node with an odd number of edges you have to finish off it; if you start off it you have to finish on it.
So we have a set of rules that logic (or Euler) tells us is irrefutable. The table shows us where you must finish for a given set of starting conditions:
| Even number of edges | Odd number of edges | |
|---|---|---|
| Start on node | On | Off |
| Start off node | Off | On |
Now, if you have two nodes connected by two edges, we can complete the task.
We can start on and finish on node A, which agrees with the rules. We can start off and finish off node B, which also agrees with the rules.
So, in order to solve the Königsberg problem, a really important general question to ask is: “How many nodes have an even number of edges and how many have an odd number?” The answer is that there are four nodes and they all have an odd number of edges.
We know that for each of these four nodes: if you start on the node you must finish off the node and if you start off the node, you must finish on the node.
We have to choose one node on which to start, so there are three nodes from which we don’t start. If we don’t start on a given node we have to finish on that node. That means, to solve the problem, we have to finish on three different nodes. That is impossible, so the Königsberg bridge problem is unsolvable.
Euler solved the problem and it is important to note that there is not an equation or even a sum in sight. Mathematics is about logic and the equations that we often associate with it are merely formal ways of expressing that logic. So I am certainly not saying that you cannot use formulae to represent the Königsberg bridge problem but I am saying the underlying logic is more important.
In 1735 Euler presented his work to the St. Petersburg Academy of Science, crested by western-reforming Russian Czar Alexander the Great that is today the Russian Academy of Sciences.
The fact that Euler formally presented it to such an august body tells you that, of course, he recognised that this was no longer about bridges but was an entirely new branch of mathematics. Subsequently other mathematicians became interested in the field and so the bridge problem gave birth to the entire field which is now called graph theory.
So how do we get from graph theory to graph databases?
Underpinning every type of database engine lies a data model; the one underpinning relational database engines is based on (to no one’s great surprise) relations. (A relation is approximately maths-speak for a table).
Now the underlying data structure is what ultimately defines the strengths and weaknesses of the database engine. Relational database engines are very good at joining tables together and then sub-setting the result by column and row. They are very bad at solving problems like the travelling salesman problem which is a classically hard computational problem to solve. In short, suppose we have 20 towns and we know the distances between every pair. A salesman has to visit each town and can do so in any order. What is the shortest route he can take?)
Graph databases are underpinned by a data model that consists of edges and nodes. You are, no doubt, well ahead of me already. In order to solve the travelling salesman problem, you can use a graph database and store the towns as nodes and the distances as edges. Graph databases are much better at solving this class of question. And they can do so much more: think about Linkedin or Facebook where people are the nodes and the “likes” are the edges. Then there’s Amazon: you are a “customer” node and products are “item” nodes. When you buy an item, a “bought” edge is created between your node and the product node.
Now the graph database engine can track out from the product node, find all the other people who bought it, see what they also bought and lo, the recommendation engine is born.
Mathematics isn’t about equations that few people can understand, it is about logic and finding patterns. It underpins all the work we do in analytics. And it’s fundamental to graph engines and graph databases. Building these systems is not merely about arithmetic: it’s about thinking conceptually and reasoning things out – often before the numbers even show up. ®
ゼロ除算はどうでしょうか：
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。
Einstein's Only Mistake: Division by Zero
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