I think you absolutely need mathematics to understand the theory of relativity.
Freeman Dyson writes when he was a child "I had read some of the popular literature about Einstein and relativity, and had found it very unsatisfying. Always when I thought I was getting close to the heart of the matter, the author would say, 'But if you really want to understand Einstein you have to understand differential equations,' or words to that effect." Later on, he goes on to say how he ordered Differential Equations by H. T. Piaggio with over seven hundred problems, most of which he solved over Christmas vacation. Then when he attempted Peter Eddington's Mathematical Theory of Relativity, it came very easily after the differential equations practice.
Peter Collier wrote his beautiful "A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity" out of a similar desire... in the introduction he mentions that as a kid he always thought relativity was easy to understand but later came to realize you really can't fully appreciate it without the necessary math. So he sat down, took a year off (if I remember correctly) and wrote this book. It's beautiful and enlightening.
To be fair, "just" reading Russian isn't that hard. A grammar textbook and a dictionary gets me through polyphonic greek lit and modern german lit. Try it! You might be surprised at how easy your brain will pick it up.
I think this is actually pretty common for graduate students of mathematics.
students are required to demonstrate the ability to read mathematics in French, German, or Russian by passing a two-hour, written language examination.
I'm considering tackling this book: "Einstein's Theory: A Rigorous Introduction for the Mathematically Untrained" by Øyvind Grøn and Arne Næss which claims to provide "an introduction to the theory of relativity and the mathematics used in its processes".
Stipulating that "understand" can cover a lot of territory, one can perhaps develop a surface and somewhat metaphorical understanding of aspects of modern physics without math. However, you're almost certainly right that any deep understanding of, say, the standard model or relativity is deeply rooted in the math.
But math is just a bunch of ideas expressed in a very efficient notation. All those notational squiggles represent ideas, and ideas can also be represented with words. It would be much less efficient, but anything explained by math should be able to be explained by words.
I had a girlfriend who insisted that "two plus two equals four" was a better expression than "2+2=4", but then she also had trouble with any concepts past algebra (and most of algebra for that matter). The notation is efficient and precise. English, and most other languages, are not. We may be able to express the idea of relativity or Newtonian mechanics or anything else in words, but the base concepts end up requiring us to move into a more expressive and precise language when we want to understand and apply (which today is mathematics expressed with the notations of algebra, trigonometry and calculus).
Math is a great example of how notation is sooo important. Doing math with roman numerals is.... possible, sure. But for pretty much any math that's worth anything, it's just a bad notation. It obscures what's going on and makes it harder to mentally deal with. Notation is a critical tool of thought.
Absolutely. The same is also true of programming, music, and natural languages. Probably nearly every discipline that uses its own notation and significantly divergent (from the primary natural language) nomenclature sees a major boost in understanding and expressivity by using custom notations and language.
Re programming: Some languages are simply better at expressing certain types of programs than others, C may be great, but it gets pretty hairy when you try to use it in certain parallel/concurrent coding contexts (i.e., you're probably using a lot of extra libraries and tooling).
Re music: Could you imagine expressing any piece of significant length or complexity in plain English? Even a mathematical (concise and precise) symbolic language would prove difficult, compared to the visual and multi-dimensional notation customarily used. (At least for reading, a case could possibly be made for more expressive and concise notations for writing musical pieces.)
Re natural languages: This is why we borrow so many words in English (and other languages, I'm sure) from elsewhere. A larger set of words allows us to produce more expressive and nuanced statements and questions.
In principle, yes, but in practice, no. Tools let us think previously unthinkable thoughts. See Bret Victor's Media for Thinking the Unthinkable [1], where he quotes al-Khwarizmi from the ninth century:
What is the square which when taken with ten of its roots will give a sum of thirty nine? Now the roots in the problem before us are ten. Therefore take five, which multiplied by itself gives twenty five, and amount you add to thirty nine to give sixty four. Having taken the square root of this which is eight, subtract from this half the roots, five leaving three. The number three represents one root of this square, which itself, of course, is nine. Nine therefore gives the square.
He's solving x^2+10x = 39 by completing the square.
You don't have to use those words. For instance, for
n >> k >> ln(n) >> 1,
he says "Specifically, we require n much greater than k much greater than log n much greater than one", and says it's unwieldy. Well, if you say it like that, then it is.
"Specifically, we require than the magnitudes of n, k, log n, and one be in decreasing order with significant distance between each."
Not a problem. Just don't try to read notation literally in the order it's written.
The Greeks did that for a while, before the efficient notation was invented. It didn't work out too well, the explanations get very complex very quickly and it's hard to hold all the facts in your head without some convenient shorthand.
Thank you for finding it. I was interested in seeing how difficult the problems were. The first page does seem it requires knowledge of calculus and partial derivatives. The problems he referred to are of the "Examples for Solution" kind like the one on page 3. For motivation, Dyson also said "The difference between a text without problems and a text with problems is like the difference between learning to read a language and learning to speak it. I intended to speak the language of Einstein, and so I worked my way through the problems. I started at six in the morning and stopped at ten in the evening, with short breaks for meals. I averaged fourteen hours a day. Never have a I enjoyed a vacation more."
Don't dismiss it so early. I'm reading it and it's very clear up to now. Just do the exercises. Anything in particular you find challenging on page one?
Comments
I think you absolutely need mathematics to understand the theory of relativity.
Freeman Dyson writes when he was a child "I had read some of the popular literature about Einstein and relativity, and had found it very unsatisfying. Always when I thought I was getting close to the heart of the matter, the author would say, 'But if you really want to understand Einstein you have to understand differential equations,' or words to that effect." Later on, he goes on to say how he ordered Differential Equations by H. T. Piaggio with over seven hundred problems, most of which he solved over Christmas vacation. Then when he attempted Peter Eddington's Mathematical Theory of Relativity, it came very easily after the differential equations practice.
Peter Collier wrote his beautiful "A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity" out of a similar desire... in the introduction he mentions that as a kid he always thought relativity was easy to understand but later came to realize you really can't fully appreciate it without the necessary math. So he sat down, took a year off (if I remember correctly) and wrote this book. It's beautiful and enlightening.
Looks awesome and at $13, probably a good buy: http://amzn.com/0957389450
If I'm not mistaken Dyson also learned Russian one summer as a kid, because he wanted to be able to read certain book. What a guy.
To be fair, "just" reading Russian isn't that hard. A grammar textbook and a dictionary gets me through polyphonic greek lit and modern german lit. Try it! You might be surprised at how easy your brain will pick it up.
I think this is actually pretty common for graduate students of mathematics.
http://www.math.harvard.edu/graduate/
I'm considering tackling this book: "Einstein's Theory: A Rigorous Introduction for the Mathematically Untrained" by Øyvind Grøn and Arne Næss which claims to provide "an introduction to the theory of relativity and the mathematics used in its processes".
This preview on Google Books [ https://books.google.com.sg/books?id=7zFalCF_LiEC&printsec=f... ] appears to show it starts from basic mathematical principles and builds on from that.
Stipulating that "understand" can cover a lot of territory, one can perhaps develop a surface and somewhat metaphorical understanding of aspects of modern physics without math. However, you're almost certainly right that any deep understanding of, say, the standard model or relativity is deeply rooted in the math.
But math is just a bunch of ideas expressed in a very efficient notation. All those notational squiggles represent ideas, and ideas can also be represented with words. It would be much less efficient, but anything explained by math should be able to be explained by words.
I had a girlfriend who insisted that "two plus two equals four" was a better expression than "2+2=4", but then she also had trouble with any concepts past algebra (and most of algebra for that matter). The notation is efficient and precise. English, and most other languages, are not. We may be able to express the idea of relativity or Newtonian mechanics or anything else in words, but the base concepts end up requiring us to move into a more expressive and precise language when we want to understand and apply (which today is mathematics expressed with the notations of algebra, trigonometry and calculus).
Math is a great example of how notation is sooo important. Doing math with roman numerals is.... possible, sure. But for pretty much any math that's worth anything, it's just a bad notation. It obscures what's going on and makes it harder to mentally deal with. Notation is a critical tool of thought.
Absolutely. The same is also true of programming, music, and natural languages. Probably nearly every discipline that uses its own notation and significantly divergent (from the primary natural language) nomenclature sees a major boost in understanding and expressivity by using custom notations and language.
Re programming: Some languages are simply better at expressing certain types of programs than others, C may be great, but it gets pretty hairy when you try to use it in certain parallel/concurrent coding contexts (i.e., you're probably using a lot of extra libraries and tooling).
Re music: Could you imagine expressing any piece of significant length or complexity in plain English? Even a mathematical (concise and precise) symbolic language would prove difficult, compared to the visual and multi-dimensional notation customarily used. (At least for reading, a case could possibly be made for more expressive and concise notations for writing musical pieces.)
Re natural languages: This is why we borrow so many words in English (and other languages, I'm sure) from elsewhere. A larger set of words allows us to produce more expressive and nuanced statements and questions.
In principle, yes, but in practice, no. Tools let us think previously unthinkable thoughts. See Bret Victor's Media for Thinking the Unthinkable [1], where he quotes al-Khwarizmi from the ninth century:
What is the square which when taken with ten of its roots will give a sum of thirty nine? Now the roots in the problem before us are ten. Therefore take five, which multiplied by itself gives twenty five, and amount you add to thirty nine to give sixty four. Having taken the square root of this which is eight, subtract from this half the roots, five leaving three. The number three represents one root of this square, which itself, of course, is nine. Nine therefore gives the square.
He's solving x^2+10x = 39 by completing the square.
[1] https://youtu.be/oUaOucZRlmE
You don't have to use those words. For instance, for
n >> k >> ln(n) >> 1,
he says "Specifically, we require n much greater than k much greater than log n much greater than one", and says it's unwieldy. Well, if you say it like that, then it is.
"Specifically, we require than the magnitudes of n, k, log n, and one be in decreasing order with significant distance between each."
Not a problem. Just don't try to read notation literally in the order it's written.
The Greeks did that for a while, before the efficient notation was invented. It didn't work out too well, the explanations get very complex very quickly and it's hard to hold all the facts in your head without some convenient shorthand.
Not just an efficient notation. Also very precise.
Natural languages are not precise, and they lead to misunderstandings.
The ideas are math, not the notation. So once you've converted the notation to words, you've just converted the math into a different representation.
It's still math.
I've given up on this book at page one:
https://archive.org/stream/DifferentialEquations_91/Piaggio-...
Thank you for finding it. I was interested in seeing how difficult the problems were. The first page does seem it requires knowledge of calculus and partial derivatives. The problems he referred to are of the "Examples for Solution" kind like the one on page 3. For motivation, Dyson also said "The difference between a text without problems and a text with problems is like the difference between learning to read a language and learning to speak it. I intended to speak the language of Einstein, and so I worked my way through the problems. I started at six in the morning and stopped at ten in the evening, with short breaks for meals. I averaged fourteen hours a day. Never have a I enjoyed a vacation more."
Don't dismiss it so early. I'm reading it and it's very clear up to now. Just do the exercises. Anything in particular you find challenging on page one?