Visiting Leibniz: A collision with a master through time and space

I've been interested in Gottfried Leibniz for years, especially since he seemed to want to create a tool like Mathematica and Wolfram Alpha 3 centuries ago and probably write a new science. So, on a recent trip to Germany, I was thrilled to be able to visit the Leibniz Museum in Hanover.

Flipping through his yellow manuscript (still quite enough to withstand my touch), I tried to imagine his thoughts in writing these chapters, trying to connect what I saw here with what we had learned 3 centuries later, and I felt a resonance.

Some of these records, especially in mathematics, transcend time, such as Zlaibnitz's infinite progression of convergence to √2 (Latin in text):

The infinite series that Leibniz wrote to converge on √2

In another example, Leibniz tried to calculate the value of the hyphen, although his algorithm was wrong, but he still recorded the entire process ("Π" is equivalent to an earlier version of the Equals sign):

Calculating the value of a continuous fraction, although the algorithm is wrong, Leibniz still records the whole process.

For example, a summary of calculus can be almost included in modern textbooks:

A summary of the calculus by Leibniz

But what else is there? What is the macro-picture of Leibniz's work and thought?

I have always felt that Leibniz's image is somewhat elusive. He did a lot of seemingly disparate and irrelevant things--involving philosophy, mathematics, theology, Law, physics, history. And the language he used to describe his work, as we see it today, is a strange wording from the 17th century.

But as I learned more and more deeply about Leibniz as a person, I sensed a core mindset that was hidden in his many accomplishments, and this direction of thinking coincided with the modern computer concept I was pursuing.

The pursuit of systematization and structure of knowledge

In 1646 (the 4th year after Galileo's death, and the 4th year after Newton's birth), Gottfried Leibniz was born in the Leipzig area, which now belongs to Germany. His father was a philosophical professor and his mother was from the book Trade family. When Leibniz was 6 years old, his father died. Considering his youth, Leibniz was allowed to enter his father's study 2 years later and began to wander through the books. He entered the local university at the age of 15 to study philosophy and law, and graduated from both disciplines at the age of 20.

Even in the year of the Annals of Learning, Leibniz seemed interested in the systematization and standardization of knowledge. There has been such a vague view of the long-term existence-for example, the 14th century Lamon Liuli (Ramon Llull, a writer, philosopher, and logic artist of the Kingdom of Mallorca [now Spain], expressed in his semi-mystical book, "TheArs Magna") That is, we can build some kind of universal system, in which the various combinations of symbols taken from an appropriate (Descartes so-called "human Thought Alphabet") may express all knowledge. In his dissertation on philosophy, Leibniz tried to explore this idea. He used some basic combinatorial mathematical knowledge to calculate the probability. He also mentions the decomposition of ideas into simple elements that can be processed using the "logic of Creation". In addition, he has joined a claim to prove the existence of God's argument.

As Leibniz said in his later years, this essay, which he wrote at the age of 20, was naïve in many ways. But I think Leibniz began his life-thinking on all kinds of problems. For example, Leibniz's thesis on legal graduation Theses is "difficult legal cases", which are discussed in this paper as a possibility to solve the problems of logic and combinatorial mathematics.

Although originally expected to be a professor, Leibniz eventually decided to spend his life advising multiple courts and rulers. Sometimes he would like to contribute his knowledge, trace the important political significance of the genealogy or history, sometimes, such as code, literature, such as systematic normative collation, and sometimes to carry out practical engineering design, such as the planning of silver mine drainage plan; there are times-especially in early years-that he has to provide for political initiatives " Real-time on-the-ground "intellectual assistance.

In one such political action in 1672, Leibniz was sent to Paris, where he spent 4 years--during which time he met many of the leading academics of the day. Prior to this, Leibniz's mathematical knowledge was only at the basic level. But in Paris, he had the opportunity to learn all the most advanced ideas and methods. For example, he had found Christiaan Huygens and successfully passed the test-the sum of the reciprocal of all the triangles-and the latter agreed to instruct Leibniz to learn maths.

After years of hard work, Leibniz perfected his theory of systematization and standardization of knowledge, and has been conceiving a whole structure that can make knowledge--according to the present theory--computable. His first step was to develop a semiotic (ars characteristica)-A methodological study of the use of symbols to represent things, and to actually develop a unified "mind Alphabet". In his next vision, through this set of single-fingered systems, it is possible for us to "find the reasoning truth in any field by calculus [1], as in arithmetic and algebra." "This has a surprisingly common denominator with the computational theory we know today.

He used a lot of ambitious ideas when referring to his philosophy, such as "general knowledge Approach", "Philosophical Language", "Universal Mathematics", "Universal system", and "thinking algorithm". He expects the system to eventually be applied in all fields: Science, Law, medicine, engineering, theology, and so on. But in one of the subjects, he quickly achieved remarkable achievement, that is, mathematics.

As far as I know, the mathematical history of mathematical symbols as a central subject to study the case is surprisingly rare. In just a few cases, such as the late 19th century, the beginning of modern mathematical logic, Gottlob Fregg (Gottlob Frege, German mathematician, philosopher, founder of Mathematical logic) and Giuseppe Piano (Giuseppe Peano, Italian mathematician, logic and linguist , a pioneer in mathematical logic) and other people's work. And some of my attempts to build Mathematica and the Wolfram language in recent years. But Leibniz began the work 3 centuries ago. And as far as I can guess, Leibniz's achievements in the field of mathematics owe much to his efforts in the symbolic system and to the clearer mathematical structure and process of the system.

Achievements in the field of mathematical symbology

When we read Levin's paper, we find that the symbols he uses and their evolution are fascinating. Many of them look very modern. Although there are also a few 17th century scribblings, for example, he occasionally uses alchemy or astrology symbols to represent variables in algebra:

Leibniz represents variables in algebra by alchemy or astrology notation

Here he uses π as an equal sign and slightly threadbare as a balance: the "leg" of one side is written slightly longer to indicate less than ("<") or greater Than (">"):

Leibniz represents greater than ("<") or less than (">")

The underscore here is used to denote the merging of similar terms--a better idea than parentheses, although it is inconvenient to type and typesetting:

Leibniz uses an underscore to represent the merging of similar terms

Today, we are going to use the root of the radical. But Leibniz wanted to use the symbol in his points, with a "D" with a pretty little tail. It reminds me that we use the blackboard bold "differential D" in Mathematica to represent integrals.

Leibniz uses the symbol of the root in the integral

± is often used when solving equations, but this often makes the grouping process very confusing, such as a±b±c. Leibniz seems to have had a similar problem, but he invented a notation to solve the problem-a method that is worth using even now:

Leibniz invention Mark grouping ±

Some of the marks used by Les let me be unclear. But these on-wave lines are really pleasing to the eye:

And these little dots:

Or a chart that looks interesting:

Of course, Leibniz's most famous symbol is to count the integral symbols he created (with long "S" for "sum") and "D". The system was first summed up in the blank space of the paper, dated November 11, 1675 ("1675" in the Aftermath "5" was changed to "3", perhaps from Leibniz's handwriting):

The integral symbol created by Leibniz

The interesting thing I noticed was that, despite the creation of these "mathematical" arithmetic symbols, Leibniz apparently did not invent a set of similar symbols for logical operations. "or" just using Latin "vel" means "and" is "et", and so on. When he thought of the idea of logical quantifiers (such as modern ∀ and ∃), he was only using the Latin abbreviation U.A. and P.A. Quickie.

Leibniz used the Latin abbreviation U.A. and P.A.

An attempt to create an "arithmetic machine" early

What has always made me feel abnormal is that in the history of thought, the concept of the generalized generalization operation (Universal computation) didn't sprout until the 1930s. And I always wondered if Leibniz's manuscript contained an earlier version of the generalization operation--perhaps even a pattern for modern-day interpretation of a Turing-like system. But with the increasingly profound contact with Leibniz, I clearly see why this is not the case.

One of the important reasons, I presume, is that he does not pay enough attention to discrete systems. He described the results of combinatorial mathematics as "self-evident", presumably because he considered that the results could be directly demonstrated by computational methods. For him, only "geometrical" or continuous mathematical problems are worth inventing calculus to solve. When describing problems such as curve characteristics, her came up with a method similar to continuous function. But he never applied this idea of function in discrete mathematics-and that would probably lead him to think about the common elements of building functions.

Leibniz recognized the success of his calculus and was bent on creating similar "calculus" for other areas. In his experience of another miss of the generalization operation, Leibniz thought of using numbers to encode logical features. He conceived that each possible nature of something should correspond to a different prime number and then describe it by multiplying the number of prime numbers representing its nature-and then substituting mathematical operations for the logical deduction process. But he only considered the static nature--and never thought of it as a number of numbers to encode the operation.

Although Leibniz did not have the idea of generalization, he indeed the idea that computing is, in a sense, mechanized. And he seems to have made up his mind in the early days to build a real mechanical computer for mathematical operations. Probably partly because of his own convenience (this is a panacea for developing new technologies!). Because, aside from his accomplishments in algebra and other aspects, his manuscript is filled with basic (and some wrong) formulas-and these are preserved for the future:

In Leibniz's time, there had been sporadic instances of the construction of mechanical computers, and in Paris he had undoubtedly seen the addition calculators built by Pascal in 1642. But Leibniz is committed to building an "all-rounder" computer, which will be the first to perform all 4 basic operations on a single machine. He also wants to design a simple "user interface" for the machine: the operator can pull the handle to the side to multiply, and the opposite direction is the division operation.

In Leibniz's manuscript, a variety of diagrams that explore the workings of the machine are everywhere:

Various diagrams in Leibniz's manuscripts

Leibniz had imagined that his computer would have a good real-world function-in fact he seemed to want to develop it into a successful business. But in fact, it's hard for Leibniz to make this computer work stably. Because like other mechanical computers of that era, the machine was just an exaggerated odometer. Similar to the machines of Charles Babbage (Charles Babbage, British mathematician, inventor, and mechanical engineer) nearly 200 years later, it is difficult to achieve a large number of rotary tables at the same time from a mechanical point of view when large-scale movements occur.

Leibniz initially built a wooden prototype that was planned to handle only 3 to 4 digits of arithmetic. But during his 1673 visit to London, the prototype was not as good as it was shown to Robert Hooke and others. But he always thinks he can solve all the problems-for example, he wrote in 1679 (in French) the "Arithmetic Machine final amendment":

The final amendment to the arithmetic machine, written in Leibniz's 1679 usage text

However, a 1682-year note notes that there are more questions to be solved:

But Leibniz still drafted a plan based on his notes-and contracted an engineer to build a copper version capable of handling higher numbers:

Reading the "Marketing Materials" written by Leibniz for this machine is an interesting thing:

Leibniz writes "Marketing material" for "Arithmetic machine"

There are also some "use Notes" (with 365x24 calculation as "work samples"):

Instructions for use of the "Arithmetic machine"

and attached with a detail of the use of the knot:

A detailed use of the "Arithmetic machine"

Despite all the effort, the problems of the calculator have not been solved. In fact, for more than more than 40 years, Leibniz has been insisting on debugging his calculators-about a total of more than $1 million.

So what is the final physical whereabouts of this computer? The question was raised when I visited the Calculus Museum. "Well," said the host, "You can have a look." "In a storeroom, between the shelves of boxes, Leibniz's calculator was placed in a glass box that looked brand new--and I took this picture of the ancient and modern weird collocated:

Leibniz created the "calculator", background looming the author, is taking pictures of the Wolfram.

All the parts are here. Includes a portable wooden storage box. A crank handle is also provided. In addition, if everything works, a few minutes will give it the ability to handle all basic mathematical operations:

Leibniz invented the hand-cranked computer details that can do arithmetic.

The essence of number and arithmetic: Leibniz and 2 binary

Leibniz clearly regarded his computer as a practical project. But he still hopes to draw some conclusions, such as a universal "logic" that can be used to describe mechanical linkage geometry. At the same time, he pondered the nature of numbers and arithmetic. And a way to come up with 2 binary.

For centuries, a carry system other than 10 binary has been applied to interesting mathematics. But Leibniz believes that the 2 system has a special meaning-perhaps it is an important hub for connecting philosophy, theology and mathematics. After he communicated with the missionaries who had returned from China, and realized that the 2 system was the core idea of the book of the Ching, there was a greater motivation and thought that it was similar to his own "universal systems".

Leibniz wondered about the possibility of building a 2-based computer. But he still seems to think that only 10 of the system has practical significance.

Leibniz's reading of the 2-binary record is a bit strange. Some parts are clear and practical--and still seem very modern. But there are parts that are very much in the 17th century--such as discussion 2, which proves that everything comes from nothingness, 1 of which can be regarded as God, and 0 symbolizes none.

For several centuries after Leibniz, few people had made a few gains with 2: In fact, the rise of digital computers in recent decades had changed the situation. So, look at the Levin manuscript, where his calculations in 2 are probably the most "Beyond The Times":

The calculation of the binary system in Leibniz's manuscript

Through a 2-based study, Leibniz explores the simplest possible infrastructure in a sense. There is no doubt that he is doing a similar job when discussing what he calls the "list" concept. I have to admit, I never really understood the list. Whenever I think I'm going to figure it out, the part that mentions the soul is always confusing.

Nonetheless, Leibniz seems to have inferred that "the best of all possible worlds" is the one that "builds the most diverse phenomena by the least rules", which has always fascinated me. In fact, before writing a new science, it was 1981, and I was just starting to learn and build a cellular automaton, and I thought about naming them "cluster (polymones)"--but at the last minute, the list theory once again confused me and scared me back.

Documents and manuscripts for storage

Leibniz and his documents have been wrapped in a mysterious layer of color. Courtes Godel-perhaps his paranoia-seemed to have believed that Leibniz had discovered the great truths that had been suppressed for centuries. But although his manuscript was indeed sealed after Leibniz's death, it was due to his research in history and genealogy-and the state secrets that might be involved.

Leibniz's documents were opened a long time ago, and 3 centuries later we may think that all aspects have been thoroughly studied. But the truth is, even in such a long time, no one has ever really browse through all the manuscripts in detail. This is not because the file size is too large. All of these files come in a total of just 200,000 pages--an estimated more than 10 spaces on the shelves (which is only slightly larger than my personal documents since 1980). The real problem is the diversification of materials. Not only involves a variety of disciplines. And because there are a lot of overlapping drafts, notes and letters, the relationship is not very clear.

The Leibniz Museum of Literature holds a series of puzzling documents. From a huge size:

To very mini (with age, myopia becomes more serious, Leibniz's words are also smaller):

As he grew older, Leibniz's words were smaller and more written.

Most of the files in the file look very serious and cautious. But despite the price of paper in that era, we can still find the her of the graffiti remains (will this be Spinoza?). ）：

Leibniz had a correspondence with hundreds of people--both celebrities and ordinary--with letterhead throughout Europe. 300 years later, the future generations can find Jacob Bernoulli and other people sent the "essay Short note":

Jacob Bernoulli and others sent Leibniz's "essay note"

What does Leibniz look like? See here, with his official portrait, and the version without the oversized wig (even at the time), presumably he did it to cover a large cyst on his head:

Leibniz, it is said that he wears a wig to hide the cyst on his head.

In the Leibniz literature Hall, in addition to a large number of documents and his mechanical computer, there is also an item: He went out with a folding chairs around him, he hung it in the car, so he can continue to write when the car is moving:

Leibniz goes out with a folding chairs on his side, allowing it to continue writing when the car is moving.

We can not help but wonder what kind of proverbs are engraved on Leibniz's tombstone. But in fact, when Leibniz died at the age of 70, his political career had fallen into the doldrums, and no man had built a beautiful memorial hall for him. Still, I was very eager to see his tomb when I was in Hanover-but I found that the monument was simply written in Latin: "Leibniz's bone-burying place."

Leibniz's tombstone, which simply wrote "Leibniz's bone-burying place."

However, at the other end of the city, I found another form of remembrance--the biscuit in a direct selling shop was named Leibniz's name to show his respect:

A cookie named for Leibniz

Leibniz, the limit under achievement

So, in the final analysis, what should we think of Leibniz? If history develops in another form, perhaps Leibniz will establish a direct connection with modern computer technology. But the fact is that most of Leibniz's attempts are isolated-to understand that his work relies heavily on projecting modern computer theory back into the 17th century.

With what we now know, it is easy to see what Leibniz has mastered and what he has failed to understand. He understood the concept of using standardized, symbolic indicators to represent many different things. He also speculated that there might be generalized elements (perhaps even just 0 and 1) that could be used to form these indicators. And he realized that, from the normalization and symbolic representation of the knowledge, it was possible to calculate the results mechanically-perhaps through all the possibilities of poverty-to open up new knowledge.

Some of Leibniz's records appear too abstract and metaphysical-sometimes irritating. But in a way, he is quite pragmatic. And he is technically capable enough to make real progress. His usual approach seems to be to start with an attempt to create a normative structure to clarify things--and, if you can, use the symbols of the norm. After that, his goal is to create an "algorithm" that can be systematically drawn to conclusions.

To be honest, he only succeeded in a particular field with this approach: continuous "geometry" mathematics. It is a pity that he never studied in discrete mathematics. Because I think he might be able to make some progress, and it's not even hard to imagine the idea of a generalization operation. He might end up enumerating the possible systems, as I did in the computer field.

He also experimented with this approach in another field, which was law. But he started too early in this direction, until now--300 years later--computational law has just begun to show practical significance.

Leibniz also made an attempt in physics. But although he has achieved some concrete concepts (such as kinetic energy), he has never been able to summarize a large "world system", as Newton did in his "principles" (in this case, "the Mathematical Principles of Natural Philosophy").

In a way, I think that Leibniz did not achieve a higher accomplishment because he was too obsessed with practicality, and--This is like Newton--to deconstruct the physical process, rather than focusing on the structure of the relevant form. Because if Leibniz had at least tried some of the basic explorations I made in "a new science"--I don't think it was technically difficult for him--then the history of science would have to be rewritten.

I also began to realize that when Leibniz lost to Newton in the public relations War of calculus, it was not only his personal reputation that threatened him, but the way of thinking about science. Newton was a typical pragmatist in a sense: he invented a tool and then showed how to apply it to the practical problems of the computational material world. But Leibniz's vision is broader and more philosophical, and he argues that the essence of calculus is not a tool, but a paradigm that motivates us to study other areas of normalization and other universal tools.

I often think that the modern computational thinking mode that I pursue is a characteristic that is obvious and inevitable in standardized and structured thinking. But I have never clearly realized whether this significance is only the result of this era, and our experience of using modern, practical computer technology. The attention to Leibniz gives us a new perspective. In fact, we can see some of the core ideas of modern computational thinking, even as far back as it is possible. However, the limitations of the technological environment and the way of understanding over the past centuries have defined the limits of the future of this thought.

This, of course, brings a sobering question to us today: how much lag is there in understanding the core of computational thinking because it does not have the future of science and technology? For me, the study of Leibniz made me more focused on this issue. And one thing that I can clearly foresee.

In Leibniz's life, he had seen few computers, and they could only do basic mathematical operations. There are now 1 billion computers in the world, and they are capable of all kinds of jobs. But in the future, the number of computers must be much larger than this (the computer will be easier to manufacture by the computational equivalence principle). And there is no doubt that all of the items we produce will obviously be made by computers at all levels. In the end, everything must be programmable, small to atomic. Of course, biology has achieved this in some way, but there are still many constraints. But we will be able to do it completely in the future, no matter where.

To some extent, we can already see that this implies a partial combination of the computational process with the physical process. But for us, the difficulty of speculating about this integration is like making Leibniz imagine Mathematica and Wolfram Alpha.

Leibniz died on November 16, 1716. By 2016 it would have been 300 years. We can make use of this great opportunity to ensure that we are finally able to thoroughly study the entire legacy that Leibniz has left us-and to celebrate how many important visions of Leibniz have become a reality 3 of a century later, even in a way that he could never have imagined.

Content comments

[1] Leibniz divides the truth into reasoning truth and factual truth. The reasoning truth is universal and inevitable, the contradiction law of logic can deduce it, its opposite is impossible. The truth of fact is accidental, it is derived from inductive, must conform to the law of sufficient reason, its opposite is possible.

http://www.guokr.com/article/437228/

Visiting Leibniz: A collision with a master through time and space