Aristotle pioneered the logical system, which is good, but ...
Leibniz put forward binary arithmetic with only 0 and 1, which became the operation mode of modern computers. However, modern computers are no longer calculators, that is, adding, subtracting, multiplying and dividing by the number of revolutions of gears, but by the switching state of electronic components. How to do binary operation on the switch state? This requires the help of "logical operation".
The word logical operation sounds natural in modern times, but if you ask people before the eighteenth century, they will be puzzled. Logic uses words and sentences, not mathematics. How to operate?
Yes, since Aristotle created a logical system with rules to follow, the way to express logic for two thousand years has been to use natural language as a declarative sentence, such as the following most representative syllogism.
Major premise: everyone will die.
Minor premise: Socrates is human.
Conclusion: Socrates will die.
Aristotle statue
However, natural language will inevitably have multiple meanings or ambiguities. Not only translation into different languages may cause misunderstanding, but even people who use the same language may have different understandings of logical relations. For example, the word "relationship" in the proverb "it doesn't matter if there is a relationship, it doesn't matter if there is a relationship" obviously has different meanings, and the usage of "you" and "no" is also inconsistent.
In addition to simple syllogism, there are other forms of more complex logical statements, and it is really impossible to accurately express various logical forms and laws in natural language.
Do you think all mathematical formulas are like this?
In fact, in the early days of mathematics, natural language was also used. If you open the math books at that time, you will only see a long narrative. Even if you can understand it, I'm afraid it's hard to relate it to a simple formula. This is because it was not until the sixteenth century that mathematics began to be represented by symbols, such as addition, subtraction, multiplication, division and equal sign, and it was about that time that it was changed to+,-,×, ⊙ 8857; And = The familiar mathematical symbols, including using letters to represent unknowns, only became popular in the 17th century.
Addition, subtraction, multiplication, division, etc. were not changed to+,-,×, present sum = etc. until16th century.
After mathematical symbolization, the expression is shorter and more accurate, and the calculation becomes much more convenient. Moreover, because symbols are not separated by language, mathematicians from different countries can see them at a glance, so they are an important cornerstone for accelerating the spread and exchange of mathematics and science.
Leibniz himself traveled to Germany, France and Britain, and created new symbols for calculus, so he felt the importance of symbolization. Therefore, before he dreamed of owning a universal mathematical text, he had tried to transform logic into a mathematical expression, which was called "mathematical logic" in modern times.
Leibniz devoted himself to the mathematicization of logic, but unfortunately no one knew it.
Before 1679, Leibniz assumed that all basic concepts were represented by a prime number, such as "2" for "animal" and "3" for "rationality", then the sentence "man is a rational animal" is equivalent to "6=2×3", that is, the number representing "man" is ". 6÷3=2 can be deduced from 6=2×3, which means "human irrationality equals animals", so that logical reasoning can be completed through calculation.
In 1686, Leibniz changed the letter symbols such as A, B, C … to express common propositions, and introduced symbols such as "No", "Equality", "Unequal", "Ownership" and "No Ownership", and then used these symbols to list the operation rules for dealing with set relations, such as commutative law and transitive relation.
1690, he incorporated addition and subtraction into logical calculus, making the symbolization and mathematicization of logic more complete, which can be said to lay a solid foundation for mathematical logic. Unfortunately, these manuscripts were never published before Leibniz's death, and people didn't know his research in this field until the early 20th century. Therefore, the development of mathematical logic was delayed for a century and a half before it was recreated by British mathematician george boole.
1860 Portrait painted by Boolean
Boolean PR changed the theme and re-created mathematical logic.
Bull 18 15 was born in a rural town, and his father was a shoemaker. Because of his poor family, he never received formal education after graduating from primary school, but taught himself Chinese and mathematics.
When Bull was sixteen, he was hired as a teacher by a local school and became the breadwinner of his family. At the age of nineteen, I started my own school and devoted myself to mathematics research. At the age of twenty-three, Boolean began to publish mathematical papers, and gradually gained the attention of academic circles in London. Augustus de morgan, one of the mathematicians, became friends with him and later gave him a chance to start a new logic game.
/kloc-philosophers in the 20th century noticed many problems in Aristotle's syllogism, so some scholars, including Augustus de Morgan, began to think about how to mathematize logic (as mentioned above, they didn't know that Leibniz had done research). 1846, de Morgan published a paper on syllogism, mainly aimed at the quantitative discussion of "all", "some" or "most" in the proposition.
Unexpectedly, after the paper was published, another British philosopher, Sir william hamilton, immediately jumped out and accused Augustus de Morgan of plagiarism. As a good friend of De Morgan, the bull should naturally care about the content of their dispute. Unexpectedly, after in-depth study, he became a grandmaster from the initial bystander.
1847, Boolean published Mathematical Analysis of Logic. This 82-page booklet immediately shocked the philosophical and mathematical circles. Traditional logic is completely expressed in the form of algebra, with logical relations such as AND, OR, NOT and IF, and the truth value of the proposition is represented by 1 and 0. In addition, Boolean expounded some basic axioms, such as associative law and distributive law. , and successfully mathematicized the logic.
From this kind of logical reasoning, we can use simple and accurate mathematical calculation, which not only avoids the fallacy caused by ambiguity, but also greatly increases the efficiency of dealing with propositions. With the input of many scholars, the brand-new route of mathematical logic has developed rapidly, and Boolean himself has strengthened the whole system more completely in the Law of Thought published by 1854.
In fact, many of Boer's research results have been done by Leibniz, but history is so wonderful. Leibniz is regarded as the founder of binary system, because harriot didn't publish a paper a century ago. Now, Leibniz himself did not sort out and publish the research on logical algebra, but let logical algebra be named Boolean algebra a century and a half later (called Boolean algebra, and Boolean means Boolean).
Representing Logical Propositions by Boolean Algebra
It is also a pity that Leibniz values binary so much, but does not use 1 and 0 to express the authenticity of propositional operation like Boolean. This is absolutely necessary for computer operation, so from the point of view of computer development, even if Leibniz's manuscript was published earlier, Boolean will definitely write it down in the credit book for inventing the computer.
Binary and Boolean algebra are ready, but modern computers only need the east wind.
In fact, Boolean is no stranger to computers. Because her good friend Augustus de Morgan is Ada's lover's math tutor, Boolean wrote to Babbage through this relationship. In his letter to Babbage at 1862, he especially thanked him for explaining the details of the extension for himself. Just as Leibniz once imagined a binary computer, we can't help but wonder whether the history of computers will be changed if Boolean's brand-new ideas are combined with Babbage's design talent.
But there is no way to know, because Bull died two years later. It turned out that Parr braved the heavy rain to teach at school, so he caught a cold and had a fever. Unexpectedly, his wife, who was superstitious about homeopathy, continued to pour buckets of water on Parr, causing him to get severe pneumonia and die at the age of 49.
In any case, there would be no modern computer without electricity, so even if binary and Boolean algebra are ready, we still have to wait for the east wind, that is, electricity and hardware, before computers can sail to a new generation. Of course, when the east wind rises, there must be a Zhu Gekongming to plan it. "The only way to correct our reasoning is to make them as reliable as mathematicians' calculations, so that we can find mistakes at a glance. When people argue endlessly, we just need to say: stop arguing and let's find out who is right.
Aristotle pioneered the logical system, which is good, but ...
Leibniz put forward binary arithmetic with only 0 and 1, which became the operation mode of modern computers. However, modern computers are no longer calculators, that is, adding, subtracting, multiplying and dividing by the number of revolutions of gears, but by the switching state of electronic components. How to do binary operation on the switch state? This requires the help of "logical operation".
The word logical operation sounds natural in modern times, but if you ask people before the eighteenth century, they will be puzzled. Logic uses words and sentences, not mathematics. How to operate?
Yes, since Aristotle created a logical system with rules to follow, the way to express logic for two thousand years has been to use natural language as a declarative sentence, such as the following most representative syllogism.
Major premise: everyone will die.
Minor premise: Socrates is human.
Conclusion: Socrates will die.
Aristotle statue
However, natural language will inevitably have multiple meanings or ambiguities. Not only translation into different languages may cause misunderstanding, but even people who use the same language may have different understandings of logical relations. For example, the word "relationship" in the proverb "it doesn't matter if there is a relationship, it doesn't matter if there is a relationship" obviously has different meanings, and the usage of "you" and "no" is also inconsistent.
In addition to simple syllogism, there are other forms of more complex logical statements, and it is really impossible to accurately express various logical forms and laws in natural language.
Do you think all mathematical formulas are like this?
In fact, in the early days of mathematics, natural language was also used. If you open the math books at that time, you will only see a long narrative. Even if you can understand it, I'm afraid it's hard to relate it to a simple formula. This is because it was not until the sixteenth century that mathematics began to be represented by symbols, such as addition, subtraction, multiplication, division and equal sign, and it was about that time that it was changed to+,-,×, ⊙ 8857; And = The familiar mathematical symbols, including using letters to represent unknowns, only became popular in the 17th century.
Addition, subtraction, multiplication, division, etc. were not changed to+,-,×, present sum = etc. until16th century.
After mathematical symbolization, the expression is shorter and more accurate, and the calculation becomes much more convenient. Moreover, because symbols are not separated by language, mathematicians from different countries can see them at a glance, so they are an important cornerstone for accelerating the spread and exchange of mathematics and science.
Leibniz himself has traveled to Germany, France and Britain, and once created new symbols for calculus, so he can feel the importance of symbolization. Therefore, before he dreamed of owning a universal mathematical text, he had tried to transform logic into a mathematical expression, which was called "mathematical logic" in modern times.
Leibniz devoted himself to the mathematicization of logic, but unfortunately no one knew it.
Before 1679, Leibniz assumed that all basic concepts were represented by a prime number, such as "2" for "animal" and "3" for "rationality", then the sentence "man is a rational animal" is equivalent to "6=2×3", that is, the number representing "man" is ". 6÷3=2 can be deduced from 6=2×3, which means "human irrationality equals animals", so that logical reasoning can be completed through calculation.
In 1686, Leibniz changed the letter symbols such as A, B, C … to express common propositions, and introduced symbols such as "No", "Equality", "Unequal", "Ownership" and "No Ownership", and then used these symbols to list the operation rules for dealing with set relations, such as commutative law and transitive relation.
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1690, he incorporated addition and subtraction into logical calculus, making the symbolization and mathematicization of logic more complete, which can be said to lay a solid foundation for mathematical logic. Unfortunately, these manuscripts were never published before Leibniz's death, and people didn't know his research in this field until the early 20th century. Therefore, the development of mathematical logic was delayed for a century and a half before it was recreated by British mathematician george boole.
1860 Portrait painted by Boolean
Boolean PR changed the theme and re-created mathematical logic.
Bull 18 15 was born in a rural town, and his father was a shoemaker. Because of his poor family, he never received formal education after graduating from primary school, but taught himself Chinese and mathematics.
When Bull was sixteen, he was hired as a teacher by a local school and became the breadwinner of his family. At the age of nineteen, I started my own school and devoted myself to mathematics research. Bull began to publish mathematical papers at the age of 23, and gradually gained the attention of academic circles in London. Augustus de morgan, one of the mathematicians, became friends with him and later gave him a chance to start a new logic game.
/kloc-philosophers in the 20th century noticed many problems in Aristotle's syllogism, so some scholars, including Augustus de Morgan, began to think about how to mathematize logic (as mentioned above, they didn't know that Leibniz had done research). 1846, de Morgan published a paper on syllogism, mainly aimed at the quantitative discussion of "all", "some" or "most" in the proposition.
Unexpectedly, after the paper was published, another British philosopher, Sir william hamilton, immediately jumped out and accused Augustus de Morgan of plagiarism. As a good friend of De Morgan, the bull should naturally care about the content of their dispute. Unexpectedly, after in-depth study, he became a grandmaster from the initial bystander.
Boolean stepped into the field of logical mathematicization for his friends.
The truth value of the proposition is expressed by 1 and 0, and the logical relationship is transformed into mathematical operation.
1847, Boolean published Mathematical Analysis of Logic. This 82-page booklet immediately shocked the philosophical and mathematical circles. Traditional logic is completely expressed in the form of algebra, with logical relations such as AND, OR, NOT and IF, and the truth value of the proposition is represented by 1 and 0. In addition, Boolean expounded some basic axioms, such as associative law and distributive law. , and successfully mathematicized the logic.
From this kind of logical reasoning, we can use simple and accurate mathematical calculation, which not only avoids the fallacy caused by ambiguity, but also greatly increases the efficiency of dealing with propositions. With the input of many scholars, the brand-new route of mathematical logic has developed rapidly, and Boolean himself has strengthened the whole system more completely in the Law of Thought published by 1854.
In fact, many of Boer's research results have been done by Leibniz, but history is so wonderful. Leibniz is regarded as the founder of binary system because harriot didn't publish a paper a century ago. Now, Leibniz himself did not sort out and publish the research on logical algebra, but let logical algebra be named Boolean algebra a century and a half later (called Boolean algebra, and Boolean means Boolean).
Representing Logical Propositions by Boolean Algebra
It is also a pity that Leibniz values binary so much, but does not use 1 and 0 to express the authenticity of propositional operation like Boolean. This is absolutely necessary for computer operation, so from the point of view of computer development, even if Leibniz's manuscript was published earlier, Boolean will definitely write it down in the credit book for inventing the computer.
Binary and Boolean algebra are ready, but modern computers only need the east wind.
In fact, Boolean is no stranger to computers. Because her good friend Augustus de Morgan is Ada's lover's math tutor, Boolean wrote to Babbage through this relationship. In his letter to Babbage at 1862, he especially thanked him for explaining the details of the extension for himself. Just as Leibniz once imagined a binary computer, we can't help but wonder whether the history of computers will be changed if Boolean's brand-new ideas are combined with Babbage's design talent.
But there is no way to know, because Bull died two years later. It turned out that Parr braved the heavy rain to teach at school, so he caught a cold and had a fever. Unexpectedly, his wife, who was superstitious about homeopathy, continued to pour buckets of water on Parr, causing him to get severe pneumonia and die at the age of 49.
In any case, there would be no modern computer without electricity, so even if binary and Boolean algebra are ready, we still have to wait for the east wind, that is, electricity and hardware, before computers can sail to a new generation. Of course, when the east wind rises, there must be a Zhu Gekongming to plan it.