Chapter from the book "Is God a mathematician?" By Mario Livio from English: Emmanuel Lotem, Aryeh Nir Publishing
The world does not stand still. Most of the things around us are in motion, or change relentlessly. Even the Earth, which seems so solid under our feet, spins on its axis, orbits the Sun and circles (along with the Sun) the center of the Milky Way Galaxy. The air we breathe is made of billions of molecules that are all moving constantly, and randomly. At the same time, plants grow, radioactive materials decay, the temperature of the atmosphere rises and falls from day to day and from season to season, and the human life span is steadily lengthening. But this frenzy of the cosmos, in itself, did not slow down mathematics. The branch of mathematics called Differential and Integral Calculus (DIF), was founded by Newton and Leibniz specifically to enable a meticulous analysis of movement and change, in order to be able to build accurate models of them. Over the time that has passed since their days, this wonderful tool has become formidable and comprehensive. All, and it can be used to test completely different phenomena such as the movement of a space shuttle or the spread of an infectious disease As a motion picture captures movement by breaking it down into a sequence of "frames", one after the other, the Hadoua can measure changes in such a fine sieve that it allows the determination of transient quantities - such as speed, acceleration or rate of change - in a brief moment Time is like no other.
Following the giant steps taken by Newton and Leibniz, the mathematicians of the Age of Enlightenment (late seventeenth century and eighteenth century) added and expanded the Hadua into an even more formidable and multi-application field than it, the differential equations. With this new weapon, the scientists equipped themselves, who today are able to present theories Detailed mathematics of phenomena ranging from the sounds produced by a violin string to the conduction of heat, From spinning motion to the flow of liquids and gases, for a while, differential equations were the preferred tool for advancing physics.
Some of the first researchers who visited the new landscapes that opened the differential equations were the sons of the legendary Bernoulli family. From the middle of the seventeenth century to the middle of the nineteenth century, this family produced no less than eight great mathematicians. These gifted people were known there, not only for their excellent mathematics, but also - almost to the same extent - for the fierce sibling quarrels they had among themselves. All intra-Bernoullian rivalries were related to their competition for supremacy in the world of mathematics, although some of the problems over which they quarreled may not today seem extraordinarily important. Still, the solution of these complicated puzzles in many cases opened the door to the most impressive mathematical breakthroughs. All things considered, there is no doubt that the Bernoulli family played an important role in establishing mathematics as the language of many and varied physical processes.
One story will illustrate the sophistication of the minds of two of the most brilliant of the Bernoulli brothers - the brothers Jacob (1654 - 1705) and Yohan (1667 - 1748). Jacob Bernoulli was one of the pioneers of probability theory, and we will return to him later in this chapter. But in 1690, Jacob dealt with the revival of a problem that had first been wondered about, two centuries earlier, by the greatest of the Renaissance cluster men, Leonardo da Vinci: What is the shape that an elastic chain that cannot be extended would take, if it were hung between two fixed points (see Figure 31 )? Leonardo sketched several such chains in his notebooks, and Descartes also heard about the problem from his friend Isaac Beckmann, although there is no evidence that Descartes actually tried to solve it. In the end the problem got a name, the chain line problem. Galileo thought the shape would be parabolic, but the French Jesuit Ignatius Fardi (1636 – 1673) proved that he was wrong. But Fardi did not undertake the task of finding a mathematical solution to the correct form.
Only one year has passed since Yakov Bernoulli presented the problem, and already his younger brother Yohan came and solved it (with the help of a differential equation). Leibniz and the Dutch mathematical physicist Christian Huygenz (1629 – 1695) also solved it, but Huygenz's solution made use of a more complicated geometric method. Yohan's success in solving a problem that was fortified by the strength of his brother, who was also his teacher, gave the young Bernoulli so much satisfaction that even thirteen years after Yakov's death he continued to brag about it. On September 29, 1718, Johann sent a letter to the French mathematician Pierre Ramon de Montmor (1678 - 1719), and did not bother to hide his pleasure at all:
You say my brother presented this problem; True and stable, but is it because he had a solution for her? no and no. When he presented the problem according to my suggestion (because I was the first to think of it), neither of us was able to solve it; We said desperate and thought it unsolvable, until Mr. Leibniz publicly announced, in the Leipzig newspaper of 1690, p. 360, that he had managed to solve the problem but he did not tell the solution, in order to give time to other analysts; And this is what encouraged us, my brother and me, to renew our efforts.
After shamelessly assuming the owner's right for presenting the problem, Yohan continued with open joy to Id:
My brother's labor was in vain; As for me, my luck played for me, because I found the skill in me (I say and I won't brag, because why should I humble the truth?) to solve it fully... The truth is that this came to me during the research that kept sleep from my eyelids for an innocent night... But the next morning I ran happily to my brother , who was still debating in his unhappiness with this Gordian connection and raised clay in his hand, because he always thought, like Galileo, that the chain line is a parabola. stop! stop! I tell him, don't continue to agonize over your attempt to prove the identity of the chain line with the parabola, because it is completely false... and here you are surprising me with your ruling because my brother found a method to solve this problem... I will ask you, do you really think that if my brother had solved the said problem , would cancel himself for my sake and refrain from appearing among the solvers, just to give me the glory of appearing alone on the stage in the honor of the first solver, alongside Messrs. Heigentz and Leibniz?
If you needed proof that mathematicians are nothing more than human beings in the end, here it is, in this story. But the family rivalry did not detract in any way from the achievements of the Bernoulli sons. In the years after the chain line incident, Yakov, Yohan and Daniel Bernoulli (1700 - 1782) continued on their path, and besides solving other similar problems of hanging strings, they also advanced the theory of differential equations in general, and presented solutions for the movements of projectiles in a medium that opposes their passage.
The story of the chain line illustrates another aspect of the power of mathematics - even seemingly trivial physical problems have mathematical solutions. By the way, the shape of the Chain Line itself continues to impress the millions of visitors to the famous St. Louis Gateway Arch in Missouri. The Finnish-American architect Ero Saarinen (1910 – 1961) and the German-American construction engineer Hanskarl Bandel (1925 – 1993) designed this structure, which was a symbol, in a way similar to that of an inverted chain line.
The amazing success of the physical sciences in discovering mathematical laws that govern the behavior of the cosmos as a whole naturally raised the question of whether or not it is possible to find similar principles underlying biological, social and economic processes. Is mathematics only the language of nature, the mathematicians wondered, or is it also the language of human nature? And even if there are no truly universal principles, perhaps it is possible to at least use mathematical tools to create models of social behavior, to explain it in the end? At the beginning of the journey, many mathematicians were confident that "laws" based on this or that version of the Hadoua would be able to accurately predict all future events, big and small.
This was, for example, the opinion of the great mathematical physicist Pierre-Simon de Laplace (1749 – 1827). In the five volumes of his book Celestial Mechanics, Laplace provided the first solution that was almost completely complete (although only approximate) to the various movements in the solar system. Apart from that, Laplace was the man who answered a question that surpassed even the power of the giant Newton: Why is the solar system so stable? Newton thought that the planets, because of the mutual gravitational pull between them, should fall into the sun or fly away from it into the depths of space, and he had no choice but to rely on the hand of God that maintains the integrity of the solar system. Laplace held a somewhat different opinion. He preferred not to rely on the finger of God, but simply provided mathematical proof that the solar system was stable for much longer periods of time than Newton had predicted. To solve this complicated problem, Laplace introduced another mathematical formalism called the perturbation theory, which allowed him to calculate the aggregate effect of the masses of small disturbances (perturbations) in the orbit of each planet. Finally, as a highlight, Laplace presented one of the first models of a covered object of the solar system - according to his influential nebular hypothesis, the solar system was formed by the contraction of a gaseous nebula.
In light of all these impressive achievements, it is perhaps no wonder that Plass, in composing a philosophical treatise on probabilities, boldly stated:
All events, including those that, due to their unimportance, seem as if they do not follow the great laws of nature, are their necessary results, just like the rotations of the sun. Out of ignorance of the connections that link such events to the general system of the universe, some have suggested that they depend on higher causes or coincidence... We must therefore see the current state of the universe as an outcome of its previous state and the cause of the state that will follow it. If we assume for a moment an intelligence capable of embracing all the amazing forces in nature and the various states of the beings from which it is made - an intelligence broad enough to subject all these data to analysis - then it will embrace in the same formula the movements of the largest bodies in the universe with the movements of the lightest of atoms; Because for her, nothing will be uncertain, and the future as well as the past will be visible before her. The human spirit, which has succeeded in bringing astronomy to perfection, has but a faint idea of such intelligence.
To be sure, when he spoke to Plass about this supposed supreme "intelligence", he was not referring to God. Unlike Newton and Descartes, Place was not religious at all. When he gave a copy of his book Celestial Mechanics to Napoleon Bonaparte, the emperor (who had heard that there was no mention of God in this work) remarked: "Monsieur Laplace, I have heard that you wrote this great book on the system of the universe and you never once mentioned its creator." Laplace immediately replied: "I had no need to introduce this hypothesis." The amused Napoleon told this answer to the mathematician Joseph-Louis Lagrange, who exclaimed: "Ah! This is a beautiful hypothesis; it explains many things." But the story does not end here. Laplace, when he heard about Lagrange's response, commented dryly: "The hypothesis, sir, does explain all things, but it does not allow us to predict anything. As a scholar, I must provide you with works that allow predictions."
In the twentieth century, the development of quantum mechanics - the theory of the subatomic world - proved that the expectation of finding the universe completely deterministic was too optimistic. To be honest, modern physics has proven that it is impossible to predict, even in principle, the outcome of any experiment. Instead, the theory only predicts the probabilities of obtaining different outcomes. The situation of the social sciences is doubly complicated, obviously, due to the multitude of interdependent factors, many of which are subject to great uncertainty at best. Indeed, the scholars of the seventeenth century realized quite quickly that the search for social principles, which would be universal such as Newton's law of gravity, was destined to fail from the beginning. The situation seems even more hopeless when it is necessary to wonder about what is happening in the souls of entire populations. But there were some resourceful thinkers who did not say despair, but instead developed a new treasury of innovative mathematical weapons - statistics and probability theory.
The chances beyond death and taxes
The English writer Daniel Defoe (1660 - 1731), best known for his book Robinson Crusoe, also wrote a work on the supernatural called The Political History of Satan. Defoe, who saw evidence of Satan's actions from all sides, wrote in this book: "In certain things, such as death and taxes, one can believe in a valid Mishna." It seems that the American statesman and scientist Benjamin Franklin (1706 – 1790) held a similar view on certainty. In a letter he wrote at the age of eighty-three to the French physicist Jean-Baptiste Leroy, it was said: "Our constitution has practically come into operation. Everything promises, as it were, that it will last; but in this world nothing can be said to be certain, except death and taxes." And indeed, it seems that the course of our lives is not at all predictable, exposed to natural disasters, sensitive to human error and in spite of random chance. Not for nothing do we give chance a finger and even a hand, not to mention blindness; Such expressions are intended precisely to express the fact that we are exposed to the unexpected, and are unable to control fate. Despite these setbacks, and perhaps precisely because of them, mathematicians, social scientists and biologists, since the sixteenth century, have been trying to seriously and methodically deal with uncertainty. Following the founding of the field of statistical mechanics, and in view of the recognition that the very foundations of physics - that is, quantum mechanics - are based on uncertainty, the physicists of the twentieth and twenty-first centuries also joined the struggle, with great enthusiasm. The weapon that researchers use in their fight against the lack of exact determinism is the ability to calculate the probability of receiving a certain result. Since there is no possibility to predict an actual result, there is no choice but to calculate the probabilities of different results. The tools that were developed to advance us beyond speculation and guesswork in science - statistics and probability theory - create the infrastructure not only for a significant part of modern science, but also for a large variety of social activities, from economics to sports.
We all use probability and statistics in almost every decision we make, even if we do so unknowingly at times. For example, you may not know that the number of people killed in traffic accidents in the United States was 42,636 in 2004. But if the number was 3 million, I'm sure you would know it. Moreover, this knowledge would surely have motivated you to think carefully before getting into the car in the morning. Why do we draw from these exact data a degree of confidence in our decision to travel by car? As we will see shortly, a key element in their reliability is the fact that they are based on very large numbers. The number of deaths in 1969 in Perio Town, Texas, whose entire population was 49 at the time, would not have convinced us to the same extent. Probability and statistics are the most important arrows in the quiver of economics, political consultants, geneticists, insurance companies and anyone who tries to draw significant conclusions from large amounts of data. When we talk about the fact that mathematics also spread to fields that were not previously included in the umbrella called exact sciences, it almost always did so through the doors opened before it by the theory of probability and statistics. How were these fertile fields born?
Statistics - the word is taken from Italian: stato means "state", and statista is someone who deals with state affairs - originally referred to the mere collection of facts by government officials. The first important work in statistics in the modern sense of the word was done by an unusual researcher - a shopkeeper in London in the seventeenth century. John Gronet (1620 – 1674) was trained in the sale of crackers: buttons, needles and even curtains. Since his work left him plenty of free time, Geront learned Latin and French on his own, and became interested in the death rolls - weekly notices of the number of deaths, according to church congregations - that had been published in London since 1604.
The publication of these reports was founded first and foremost to provide early warning of signs of the outbreak of deadly epidemics. Relying on these raw numbers, Geront began to notice interesting distinctions, which eventually led to the publication of an eighty-five-page book called Natural and Political Observations mentioned in the Delkman Key, based on the death certificates. In Figure 32 you will find an example of a table from Geront's book, which lists in alphabetical order no less than sixty-three causes of illness and death. In the dedication addressed to the president of the Royal Society, Geront stated that since his work concerns "climates, countries, seasons, fertility, health, illness, longevity and the relationship between marriages and between the ages of mankind," this is essentially a treatise on the wisdom of nature. Indeed, Geront was not content only In collecting and presenting the data, for example, based on an examination of the average number of baptisms and burials of males and females in the rural community of Romsey in Hampshire, Geront proved for the first time the stability of the marriage ratio at birth. In more detail, he found that thirteen females were born in London - ten males, and Bromzi - fifteen females for every sixteen males.
It is worth noting that he added and expressed, with an impressive Nold vision, his ambition that "tourists will ask if this is the case in other countries as well." And he added that "it is a blessing for humanity, because in this excess of males there is a natural barrier to polygamy: because in this situation women will not be able to live in the same numerical ratio and with the same equality of expenses with their husbands, when they live here and now." Nowadays, the accepted ratio between newborns and newborns is approximately 1.05. The traditional explanation for this excess of males is that Mother Nature tipped the scales in favor of male births because males, as fetuses and infants, are slightly less robust than females. By the way, for reasons that are not entirely clear, the proportion of male children has been decreasing somewhat every year, both in the United States and in Japan, since the seventies of the last century.
51 תגובות
You don't seem to distinguish between subject and object. If there is something that makes us call it an electron, it doesn't matter if the cat calls it an electron. It is not our knowledge that establishes the electron for existence (we agreed on this point that existence is not dependent), after all we do not know what an electron is, we know how to define a phenomenon that we call an electron, but in disguise we actually mean the same object that caused us to experience this phenomenon of an electron, we do not know what It is the same object because it is outside of us (science is based on refutations).
And in the same way, if thinking is something objective, it would have to be independent of some kind of consciousness. But on the other hand we see that the thinker is the one who thinks real thoughts. That is, the thought does depend on the thinker, and the thinker is aware of the very thought and not of some marginal phenomenon of thinking about (=God).
Thinking cannot be wrong, thinking can only be considered (that someone thinks it). Wrong/not wrong are just labels we attach to specific thoughts according to rules we set for ourselves.
The Hamiltonian:
I think your whole entry into the subject is condescending.
Know that there are many more things I could have written and didn't. I was content to state facts that anyone can check.
I did this in response to a question and you - instead of being silent and letting the discussion remain matter-of-fact - with small deviations here and there as a result of the fact that we are all human - decided to turn the discussion into a discussion about the question of what modesty is.
Well - in my opinion - the highest expression of immodesty is the preaching of morality.
point:
First of all - the existence of thinking is objective even if not everyone is aware of it - just as the existence of the electron is objective (and in your opinion - go explain the electron to the cat).
Besides - I didn't talk about it at all - I don't presuppose the existence of mathematics with the existence of thinking (after all, I claimed that it existed even without humans).
There is a huge difference between logic and mathematics and thinking, and one of the differences (it is difficult to depend on the logic of thinking which is the property of thinking only) is that thinking can be wrong and logic cannot.
Michael, do you think there is a toe pain in the universe that does not depend on a person to feel it when they step on it?
Look, although the laws of conservation of matter apply in a certain sense and the same applies to logical thinking. And so it seems that both of them can be said to exist. But the existence of the moon is objective (a cat also interacts with the moon) and the existence of thinking is subjective (go explain relativity to cats). To say that the subject exists like the object is against all the style of scientific thinking I know.
Michael Rothschild:
If you were to read the same thing that someone else wrote, you would think that it is a drop but a small drop a little bit.... fill in the title yourself.
If you are serious don't make fun of yourself.
And in my opinion, the chemistry student and my friends: he is just some kid who tries to lure you into falling into the net and you fall.
For commenter 45 who is not worth mentioning, people like Michael Rothschild, the editor's father, Ran Levy should be included
For TV shows!! I'm quite tired of all the empty celebs who fill all the TV programs and there are always
Boring articles about what they ate, what they drank, what they wore that blow up the websites,
There is a fairly large percentage of people with some sense like students and academics and interested people etc who at least enjoy a site like this.
And to Michael Rothschild, do not refer to commenter 45 and you are truly amazing with all your knowledge!!
student:
See what I meant?
You ask me a question, I answer honestly and some people find it painful.
That's why I really hesitated whether to answer but so be it.
Michael Rothschild:
A little modesty
student:
Thank you for your kind words.
I am an IDF pensioner and also a high-tech pensioner.
My official background is in math and computer science but ever since I made up my mind, the thing that fascinated me the most was trying to "understand the world".
It seems to me that the second part of this response:
https://www.hayadan.org.il/meta-beuty-2911082/#comment-144121
Not a bad description of "what moves me".
My knowledge in many and varied fields is probably a consequence of this motivation.
Of course, motivation alone is probably not enough, so I allow myself to think that I was also gifted with a certain talent.
This talent is also reflected - both in the years of my long service in the army (in addition to being promoted in the ranks, I won the ILA award twice and once in the Kashrah award for creative thinking), both in my academic studies (for example, I completed my master's degree in computer science in one year with high honors) And in the fact that the patents I sold freed me completely from the burden of livelihood.
As you must have seen, I try to influence other people as well - something that often drags me into mud skirmishes.
I have an acute allergy to expressions of dishonesty and when I encounter such - I do not hold back.
If I am convinced that someone is lying (saying something he does not know or even something he knows is not true) I tell him that this is my opinion. If someone expresses baseless disdain for others - I find it appropriate to let him taste the same reason. People scold me for this from time to time but I think it's the right way to drive.
There is a concept called "altruistic punishment" - the same phenomenon that encourages us to punish criminals even at the expense of discomfort or even risk to ourselves. Apparently my behavior regarding expressions of dishonesty is an expression of this trait and indeed the discomfort this causes me is great. It is not easy to deal with people who I already knew were not honest.
Therefore - encouraging comments like yours are very important to me. It's a rewarding landing in a series of frustrating wars.
I don't have a website.
Again - thank you from the bottom of my heart!
Michael
To Michael Rothschild! You should be given a TV show, you know how to answer any topic!
Thanks for replying 34 for me
We really appreciate that you answer and make an effort to answer! In almost every article that appears!!
Everyone learns from your comments and answers, many of the friends and surely many more people.
What field are you from? Tell us a little about yourself. Do you have a website?
Ail.A:
It's just not true.
When I say one and another are two I am simply using a shortened method of expression.
In modulo 2 you can work both in our world and in it, modulo 2, one and one more are zero.
Did you really think that I didn't think in advance that there would be someone who would jump in with this reservation?
Did you really think she was right?
Yair Shimron:
Does it bother you that justice prevails?
I explained things very well and I will try to formulate them for you in another form.
The existence of physics and chemistry in our world is part of the description of our world.
This is what defines the starting point from which mathematics can be used to draw conclusions.
It is impossible to conclude from mathematical considerations alone that there will be chemistry and physics.
As I mentioned - mathematics deals with all possible worlds and in order to be able to use it to draw conclusions about your specific world you need to give it data about this world.
Requiring her to give you predictions about something you are not ready to tell her is funny.
Regarding the language - I have already explained that mathematics cannot work without data.
Beyond that - there is the problem of the lack of determinism.
I don't know what exactly you are trying to say at this point because - see it's a miracle - what cannot be deduced through mathematics - cannot be deduced in any other way.
Michael,
I would like to refer to your claim from comment 39:..."In my opinion, a world in which one and one more would not be two..." is not possible.
Before that, full disclosure - I only read the article and comment 39.
A world where mathematics is based on the field of numbers of size 2 (the so-called Z2, contains {1,0}). 1+1=0.
What to do, but it is strictly legal. All the same algebraic axioms on which we base all our mathematics are also used by him.
Michael, do you always have to end with a small victory? What exactly is that starting point that I didn't reveal to you? As far as I know there are no axioms regarding the description of the language.
And what exactly is a little funny about the fact that there are no differences of opinion between us about the indirect contribution of mathematics to archaeology?
And finally, the claim that you can predict with great certainty that ten will not represent nine: but could you predict with a mathematical tool that a word or phrase would change its meaning in any way? There are aspects of language that tools - mainly statistical - can tell something. But the essence of language is the creation of meanings, and for this there is no effective mathematical tool.
Eddie:
The identity in mathematics is perfect and any discrepancy is the result of a mistake by one of the parties - a mistake in which the other party can convince him without difficulty.
And why call mathematics an expression of reason if it is not correct? Is it such great wisdom to believe a false thing?
In my opinion, it is not possible for a world in which one and another one will not be two, and whoever God is - there will be no escape from this fact either.
Michael Rothschild:
This God has no reason and no motive to bring all mathematicians to the same conclusions and has no reason to cause them any illusion.
This God creates - among other things intelligence (or more precisely - a collection of intelligences of subjects), as an integral part of the whole of creation, if also unique to a certain extent. This reason (any individual reason) is in some relation of conformity to the Creator and creation. One of the ('wonderful') results is this or that mathematical theory, and its specific adaptation to some segment of reality. That's all. Mathematics is an intelligent creation, the result of the attitude of reason to the surrounding objects, and not a realistic entity.
I did not claim that all cultures, in general, converge to the same culture - and I agree with you on this matter. Between different cultures there are usually parallels or cultural similes, and only sometimes identities in one or another cultural product. It is possible that in the future the cultures will converge into a more or less uniform superculture (if the Iranian/Pakistani/North Korean nuclear or a terrorist or a mad statesman strikes or an outbreak A nearby dimple or a huge meteor won't spoil the way).
In mathematics, the level of identity is relatively high (admittedly not perfect) - this does not mean that there is something here that has a realistic existence - but it stems from the very pattern of mathematics.
I noticed that the rhetorical question in the last sentence of my previous response is exactly a mirror image of the question the cat asked Aliza.
It makes sense because they were in the land of the mirror and we are in the land opposite the mirror.
thrower
Yair Shimron:
As the article mentions - in reality there are areas where the world does not behave deterministically.
The contribution of mathematics to the treatment of these areas is the theory of probability.
Of course, before one can apply mathematics to predict something, a sufficiently exhaustive knowledge of the starting point is necessary.
All the examples you described have a problem at this point.
This is the reason why our ability to use mathematics for predictive purposes (which is different from the predictive ability of mathematics) in these areas is limited, in principle, only to probability calculations and practically - even less than that.
It can, for example, be said with a fairly high degree of certainty that the word ten will never be used to represent the number nine.
It is a little funny to demand from mathematics the ability to predict a world where chemistry and physics work without the help of chemistry and physics.
After all, mathematics explores all possible worlds and in a world where chemistry and physics are different or do not exist, mathematics should lead us to a different prediction!
Mathematics is a way of drawing conclusions from a collection of axioms that are supposed to represent the specific world that one wants to explore. If you are not willing to reveal to me the starting point - how do you expect me to conclude what follows from this starting point?
There are worlds where mathematics can be of no use. For example, in the central and most important aspect of speech and language as a whole. There is no mathematical method today and there almost certainly won't be in the future that can predict or describe the development of meanings in language. For one example out of billions, let's take the simple word ten. A strictly mathematical word. About forty years ago, the meaning of the word was a certain quantity, which could be expressed in many ways, such as nine plus one, one hundred minus ninety, and many more. About thirty years ago a meaning was added to this word: excellent. We can easily understand the connection between the two meanings of the word. But if a person were asked to predict forty years ago how the word Esher would change, the chance of his success would be close to zero. It should be remembered that in retrospect, in view of the known result, it seems as if there was a chance for such a prediction. This is an "easy" case. In other metaphorical expressions the chances decrease even more and equal to zero.
The study of language is part of natural science, since speech is an essential part of man.
Mathematics has no use in other sciences either, for example in archeology (except when it is aided by physics and animals) and as far as I understand there is no chance that mathematical tools can predict and explain the changes in animals.
student:
And why is it necessary for a legislator to have laws?
In order for one and one plus one to become two, does someone have to legislate it or is it something that any legislator who does not arise will not be able to avoid?
I think the second option is the right one.
In general - if you start with the fact that there must be someone in order for there to be something, then you can never finish.
If you need a legislator for laws then you definitely need a creator for a legislator and a creator for a creator and so on. It just sounds silly.
If you try to evade and say that the legislator is eternal, then in my opinion it is much more reasonable to say that the laws are eternal.
When talking about the difference between mathematics and the natural sciences, they say that the natural sciences study the world as it is.
Mathematics explores all possible worlds (of which the world as it is is only one).
In other words - in any world - the laws of mathematics must be valid.
This is also the answer to the question regarding the adaptation of mathematics to what is happening in the world.
My argument is that in whatever world it is - mathematics must work and it will fit the description of the laws that govern it.
This is my deep feeling and I am truly convinced of it (in contrast to many of the religious people who for various reasons are only convinced of the existence of the Creator from the beginning).
Benjamin:
What you describe is indeed a real challenge and maybe really the real challenge in the news.
They don't know how to do this yet and it will take time before they know - but without a doubt they are moving in this direction.
At least in this world view there is a challenge and therefore there is progress.
The world view of belief in the God of the Gaps tends to protect the gaps so that the Creator has a place to live or in other words - stops all progress.
Eddie:
The riddle that the book tries to raise. How the coordinations between mathematical patterns and physical processes are formed.
And actually also between mathematical patterns and themselves. What is the legality that coordinates between apparently completely different issues.
Topics that emerge in completely different fields of research in mathematics and suddenly their surprising cohesion becomes clear.
The regularity that makes mathematics the way it is. The legality that creates the cohesion of the connections and the compatibility between different mathematical fields. One could say the desire to insist on deciphering the puzzle of the legality of the legality.
Michael Rothschild:
The real challenge is not to make something that looks "like" looks "like" and works roughly like other algorithms more or less.
The real challenge is deriving the mathematics by which evolution created minds.
No difference if the brain of a cockroach is a cat or a human. Or it is enough that we practically understand the software that manages a living cell in a complete biological framework. That is, with this insight we will be able to create artificial biology from a state of chemical elements to any complex system in one process.
This would be a real breakthrough. To understand the mind of evolution, to understand how it manages to create coordinations from a very small scale to a very large one.
If mathematics existed, there is a legislator for these laws.
The world is founded on these laws, so common sense says,
I believe in evolution, but this part of the math rules doesn't work for me because it means
There is a legislator.
Evolution needs to find a solution or there probably is for sure there is, so explain to me, Michael Rothschild will be among the explainers, thank you.
Eddie:
But what reason does God have to bring all the mathematicians to the same conclusions (and not over time but immediately).
What reason does he have to give them the illusion that logical reasoning has led them to their conclusions.
The only conceivable reason - even if God existed - is that he has no choice because mathematics exists and is correct even without him.
It is not true that cultures converge by themselves into the same culture. There is an influence here - sometimes violently - of one culture on another.
Year:
When you say that my link is ridiculous and you don't justify it at all - how do you expect me to answer you?
Instead of justifying your ridiculous claim, you make a ridiculous demand to present another algorithm while ridiculously ignoring the explanation I already gave to your previous ridiculous when I told him that we don't even have a mathematical model of chemistry yet.
Michael, when claims are presented that relate directly to your words, you better see that they are wrong.
Michael Rothschild,
To your post on the 22nd:
Your opinion is without a doubt a respectable opinion and reasons for it.
The whole topic is a worthy topic that can develop as a philosophical mathematical discussion - as an article. Maybe write one of these?
For my part, I tried to justify my position according to the amount of time given to me - and unfortunately these days it is not much.
I will therefore limit myself to just one factual comment: indeed, I was right in your assumption, but I cannot agree with any of the possibilities you raised (man is a machine; mathematics has an objective existence).
But note that for me there is also a third possibility implied in my response 21 in the penultimate section (God exists, God creates human reason - and here I will add: in his image). According to this possibility - it is no wonder that in the historical range mathematics comes out more or less 'uniform', just as it is no wonder that cultures tend, in long historical ranges, to reach similar or close, and sometimes even identical, cultural achievements.
And I am not using God here to fill 'gaps' - the concept of God was not created here ad hoc to justify this or that local problem.
In fact, this move may be the basis for an argument pointing to existence and/or regulative necessity and/or evidence for a creative deity, depending on how the argument is constructed. I can call it "the fat argument of mathematics"... This is similar to other arguments related to human reason - for example the 'argumentation from morality' that Kant built, and all the modern variations of this type of argumentation.
Year:
What is ridiculous is your words.
point:
In other words:
In the world we live in - we take existing things - such as materials and the laws of the sun and create new things that actually exist only after we created them while before that they existed only in potential (as a mathematical sentence that "if he does such and such, such and such results will be obtained.").
Michael, you love mathematics so much and examine reality through its simplifying prism. The link you gave showing an algorithm in evolution is just ridiculous, as much as it shows a nice work in mathematics.
You need to show an algorithm that can predict or at least describe in retrospect how a small animal turns into a huge animal with a huge nose, how a monkey turns into a human and a chimpanzee, how beetles turn into honey bees. There is no such algorithm.
And there is no algorithm that can predict or explain in retrospect the development of colloquial metaphors, for example how a waste of time becomes a waste of time and countless other expressions.
And there is much more in the world that mathematics cannot describe.
If you use the word existence as I do, you must understand that what exists even without Microsoft is the truth of the claim that "if they build computers of the type that Intel built and if they write programming as Microsoft wrote, then this programming will give the results that Microsoft's programming gives."
This is a mathematical claim and it would exist and be true even without humans.
My opinion is that you use the word exists in the way I do, only that you claim things that do not exist that they exist.
For example, you claim that mathematics existed even without a thought to conceive it (which is certainly the same sense of existence that I use).
And so I asked if you think that the software that Microsoft has developed or will develop also exists without a computer running them. Or at all any possible software that exists.
Eddie:
Of course I do not agree with your opinion and I suppose if you think about the question "how come all mathematicians reach the same mathematical conclusions?" You will have to bring as an explanation one of two explanations, both of which you reject.
The one that man is a machine and does not have the ability to choose, so everyone gets the same math.
The second is that mathematics has an objective existence.
I didn't just bring this fact up in my previous response, but I don't intend to start a long argument with you because it's clear to me that it will never end.
Benjamin,
I would like to explain my opinion in response 2.
Euclid's geometry assumes flat space. From here, as we know, several 'axioms' arise, from which Euclidean geometry is derived.
Riemann and Lubchevsky, 21 of which postulates curved spaces, from which arise various axioms and geometries - different from Euclidean geometry.
All geometries are equally 'correct' - insofar as they are cohesive, traceable and establish an internal logical integrity between their basic world picture, axioms and derived theorems. In the end, each of the geometries is a derivative of the physical world picture that mathematicians chose to base it on.
Similarly, various mathematical theories, 21 of which are logical speculations based on the perception of a physical world picture, or on a derived concept or conceptual system derived from this world picture. They also develop through conceptual 'imagination' that also has physical elements. This worldview and this logical speculation are simplifications, at one level or another, of reality.
Without a 'picture of the world' as above - there is no beginning, no existence, and no basis for any mathematics - and on the other hand - there is no 'one' mathematics and there is no 'one' mathematics that is possible, only it is correct.
Therefore all mathematics is not a realistic entity, but a purely conceptual system of concepts and links. Mathematics - all mathematics - does not 'exist' in order to be 'revealed' (or not revealed) but is a conceptual entity that is created, created by human reason in a cognitive, rational and creative process.
On the other hand, since it derives from the perception of reality (- every mathematical theory from its particular perception of reality -) and is perfected by way of abstraction and speculation - it may well be suitable for describing a certain section or sections of reality. Hence the effectiveness of various mathematical theories in describing certain segments of reality. But for exactly the same reason - certain mathematical theories are unable to describe other parts of reality. It turns out that reality will always surpass all of the given mathematics, since the perception of reality by human abilities is always limited, and even the creative ability at any given point in time does not find its full potential.
To summarize this point, a mathematical method is not a given reality, or a kind of platonic idea, but a system of conceptual concepts and connections, the fruit of reason and its powers. And in any case it is clear why a mathematical method at all suits some segment of reality, and why certain mathematical methods describe certain segments of reality, and not other segments of reality, and vice versa.
Similarly - there is no realistic entity that is a physical 'law'; The law is only what appears in our human perception and understanding as a template image of the very existence of matter and its operation. The physical substance 'behaves' the way it does not because it imposes a 'law' - but simply because it is like that in its being, and its behavior occurs as it behaves - by its very nature.
As much as mathematics is a great thing (- and indeed mathematics, for all existing and future theories, is a 'great thing' -) and it succeeds to a high degree, if not perfectly, in describing physical realities - this is because human reason is a 'great thing'. And it is such not only thanks to its mathematical creations, but also thanks to its other creations - language, art, various sciences (including physics - which is not only 'mathematics') and the entirety of human culture.
The 'wonder' is the very essence of human reason.
On the other hand, 'God' is not a mathematician. He does not need 'mathematics' to help him, and as much as he will need it - after all, he is not an absolute being, and his divine value is less and inferior, to the point that we will not need him - who needs God when we have mathematics that surpasses divinity (and this, in my opinion, is his deep intention of Prof. Livio - there is no need for God because we have mathematics). On the other hand, if God is the creator of everything, and sustains everything, then he is the creator and sustainer of reason - and she is the creator of mathematics (every theory in her time), as well as other great things - and she is the real wonder. In this context, and only in this context, reason has divine creation in it, and to this extent - mathematics, all mathematics - as well.
God is also not mathematics itself. By definition, it has to be above and beyond it and everything has a realistic or other conceptual concept. God, who is mathematics - does not have any added intrinsic value, therefore we do not need him. This is another form of idolatry - Spinozist, to which the words of Mario Livio or Tagmark are (or may be) an echo.
point:
And I stand by this claim.
I know you have a private and worthless definition of the word "existence" but I do not agree with you in this definition.
Michael, response 11 referred to what you wrote:
"Mathematics would have existed whether we discovered it or not"
Benjamin:
I read your mind from the beginning but as long as you projected a decent and serious person I treated you as such.
Now just saying that what you call my "beliefs" has already created many solutions to many problems and not by chance. This is not a monkey who accidentally tapped on the keyboard and came up with something he decided to call an evolutionary plan, but a man who understood the principle you refuse to understand and applied it to the benefit of humanity.
Now - please tell me: what exactly did they manage to build using your beliefs?
Michael Rothschild:
Guesses that look like and resemble.. are not missing you can believe or not as much as you can believe in aliens. If something looks like something else it doesn't prove anything about it being the same something else required.
But certainly satisfies those who want to believe it.
I think there was an article here that talked about believers in a religion who change their beliefs according to the mood that suits them.
Benjamin:
It is not just any algorithm and it is called evolutionary with great justice because it works on exactly the same principle as evolution.
We don't know how to do what you want yet because we don't even have a complete enough mathematical model of chemistry.
interested:
You can read something about Mario Livio here:
http://en.wikipedia.org/wiki/Mario_Livio
I read two of his books - one on the golden ratio (I no longer remember the name of the book, but it was probably simply called "The Golden Ratio") and the other - "The Language of Symmetry".
In both (especially the first) I got the impression that he is a rational person who stays away from mysticism (he quite scoffed at all the populist beliefs regarding the connection between the golden ratio and some objective measure of beauty) so my tendency at this point is to think that the word "God" is used by him as a metaphor (as Einstein who did not believe in God said " God doesn't play dice") - but I can't say for sure because in the meantime I just bought the book and haven't had time to read it yet him.
While writing the comment I "jumped" to Amazon and found there a description of the book that said exactly what I expected:
After I copied the text I found out that it is protected by copyright so I replace it withLink to the book description on Amazon:
Michael Rothschild:
It's just an algorithm they attached an evolutionary name.
My intention was to derive a mathematical legality such that it would be possible to activate a mini-evolution in a tiny bit.
Enter chemical elements as input to run the algorithm and get living cells, bacteria, etc.
and then to slightly change the algorithm and build with the inventions of the process mini-computers, etc.
But just guessing and getting something that looks similar doesn't go that far because there are tons of other algorithms that surely work much better.
Avi Blizovsky, Michael Rothschild, Yael Petar:
Can you give some background on Prof. Livio?
Why did he call the book that way? What did he want to prove?
Did he believe in evolution or God according to the title of the book?
And if mathematics exists in the first place, then what does that mean?
?
Michael, really.
You might say that Microsoft's software also exists.
Aerial concepts have no existence, they are in total some "effect", extremely complex of much simpler processes.
Laplace was wrong, the concept of God does make it possible to prophesy things.
For example, that phenomena like the Holocaust will not happen.
Benjamin:
I don't understand what you were trying to say, but evolution - not only does it work according to a mathematical principle, but it is an expression of a mathematical theorem.
That is why it is true not only in the field of biology and this is the basis for the fact that it is used in all kinds of fields.
Therefore - what you suggest they do is something that has been done for a long time.
Here's an example:
https://www.hayadan.org.il/evolution-of-an-efficient-search-algorithm-bgu-2907079/
Michael Rothschild:
So it can be assumed that evolution works according to mathematical laws.
Now it only remains to find out how to deduce from the mathematics the operation of evolution. Which are useful mathematical patterns that can be reproduced in an artificial system.
Mathematics is a great thing only because of the feeling it gives us.
This has nothing to do with the fact that it is the intellectual creation of man.
I can point to several wonderful things that are not the product of man's intelligent creation - things that the vast majority of people prefer to deal with rather than mathematics.
Mathematics seems to us to be a great thing because it is a vast field for an activity that we like to do - the activity of pattern recognition. See my articles on this matter The beauty.
Many people often make the claim that it is difficult to understand how mathematics works so well in describing nature.
The truth is that I do not share this wonder.
My feeling is that there is no other option at all.
In my opinion, there is no possibility at all for the existence of reason in a world without laws, and if there are laws - there is no possibility that mathematics cannot describe them.
My view on mathematics is platonic.
The math was there whether we discovered it or not.
This is actually the basis for the fact that different people from different cultures and different places all reach the same mathematical conclusions.
If there is something wonderful in this whole story, it lies in the fact that our minds have evolved enough to engage in mathematics, but the evolutionary explanation for this "miracle" is extremely satisfying.
On the question of God - the most extreme step taken on this issue is the step he took Tagmark who claims that in fact God is not a mathematician but is mathematics itself.
Emmanuel, how is the translation of Ray Kurzweil's book "The Singularity is Near" progressing? We have been waiting for 5 years for this book to be translated into Hebrew, is there already light at the end of the tunnel? Do you see the end?
http://www.tapuz.co.il/blog/ViewEntry.asp?EntryId=1065939
The author tries to crack the riddle of the unbelievable effectiveness of mathematics in explaining and using physical reality on its deepest layers.
Many of the great physicists, including Eugene Wigner, stood for this matter.
How does it happen that there is such a large correspondence between completely theoretical fields in mathematics and the explanation of so many important fields in physics. when apparently there is no clear connection between the things. General relativity is an example of this. Quantum chromodynamics is an example of this and countless other fields.
This matching puzzle seems so miraculous and obviously unintuitive that the author of the book associates this answer with a higher power in the name of the book.
There is actually no real explanation for the reason for this.
Mathematics was and still is a defined field within its known boundaries.
There is no guide or map that will predict how to go about breaking new barriers or solving difficult old problems. From this point of view, mathematics is a world of randomness because there is no way to predict when and in what way certain problems will be solved. This is despite all the knowledge accumulated so far. Even for the simplest problems this factor cannot be ruled out. Someone can definitely come and solve a problem that has already been solved in a certain way and solve it in a completely different way. So that you illuminate the same matter in a completely new light.
Eddie:
Hope this answers your question. Because it doesn't really matter how man-made mathematics is as you claim. There are still no answers to the miraculous correlations with physics and there is no map or guide to the future solutions.
Eddie: A person who reads in English most of the day would prefer to write in English. This is the way of the world.
For anyone interested in this lecture: it is delivered by Prof. Mario this Thursday and also live on the Internet. For details: http://astroclub.tau.ac.il
Mathematics is a great thing, but that is precisely because it is the fruit of man's rational spirit.
Mario Livio presents in his book a mystification of mathematics, and on the other hand - its 'realization'. This dual approach, apart from the fact that it contains an internal contradiction, is guilty of romanticism and philosophical naivete - which are often characteristic of scientists (including the greatest ones). The truth is that mathematics does not overlap with reality, it is only its abstraction, and reality in its entirety will always be elusive and wider than the measure of mathematics - it will always contain an element that cannot be grasped, cannot be predicted and cannot be verified. On the other hand, it is certain that mathematics is not a realistic existence, as if it were some kind of 'idea' or a group of ideas, but a conceptual existence or a group of only conceptual applications.
Thus, God is not mathematics and is not a mathematician - in fact, nothing is mathematics except mathematics, and only mathematicians are mathematicians, and mathematicians are ('only') human beings.
By the way, I'm interested to know - why does Mario Livio write all his books in English - even though he is fluent in Hebrew at the level of his mother tongue (and many of the phrases in his books are translations of Hebrew phrases), he was educated in Israel and taught in Israel?
Professor Mario Livio was a great lecturer in the Faculty of Physics at the Technion.