The problem with quantum physics is that it is hard to find a good human scale analogy for understanding it. Every analogy one might choose has a comprehensible and visible mechanism underlying it. Try trains running every 15 minutes with a standard deviation of three minutes, and you can get a wave function for trains, but there are still physical trains out on the tracks with observable positions and velocities. In quantum mechanics, trains can only be observed when they arrive at the station, and no one is exactly sure what it means to observe or arrive or exactly what a station is.
"Shut up and compute" is useful advice if you want to use quantum physics, but it is philosophically unsatisfying. People have devised more satisfying philosophical approaches, but a lot depends on what one finds satisfying. As with most of philosophy, it's more about the human mind, and less about the world in which that mind exists. The Book of Job covers some of this. Some people feel it is important to understand the "essence" of things. Others don't. I'll stop here and quote Joan Didion, “What makes Iago evil? Some people ask. I never ask.”
(I'm not very religious, but now and then I pray to god or the gods that no one finds it necessary to do a version of Othello with a tedious backstory explaining Iago's motivation and, possibly worse, has Othello and Desdemona meeting cute.)
There are no good human-scale analogies for understanding it, because it is not a human-scale phenomenon. We have evolved—biologically and culturally—to understand human-scale things, and have no ability to analogize to understand this particular non-human-scale thing...
I tend to view it as a problem in art. We understand things through metaphors. This applies to both things of an ordinary human scale things as well as those more extreme. In the early 20th century, there was a whole artistic movement groping with understanding a new world of machines. Our avant garde is still wrestling with quantum mechanics. (i think stories of time travel, alternate worlds and contingency are in this genre.)
Basically, we are not smart enough to understand things other than through grand narratives and metaphors, often about journeying through physical space and grabbing and manipulating objects. Is the only solution to ditch civilization, go back up into the trees, and hope that we can evolve into smarter creatures?
Maybe I am misreading, but this seems a little too strong.
Yes, we have evolved to understand human-scale things, so there is no guarantee that we will be able intuitively to understand quantum physics. But it may turn out that at some point in the future we do come to understand it in a way that "makes sense" to us.
Or maybe not. But I (at least) am very skeptical of "we will never understand X" claims.
After all, the position of philosophy is to be permanently dissatisfied.
Perhaps it's my background as the child of a math professor and a statistics professor, but "shut up and compute" always seemed to be a satisfying description for Newtonian mechanics just as well as it is for Quantum mechanics.
Coefficients of friction? Eh, the models predict reality, so just go with it.
There is a very nice Feynman passage on this: how you cannot explain the more fundamental in terms of analogizing it to the less fundamental, precisely because the less fundamental than itself needs to be explained...
Indeed, quantum mechanics seems to be an algorithm for computation and does a good job on the Hydrogen atom. The Hydrogen anion H+ (ionized Hydrogen molecule ground state) seems to defeat it and requires a high-powered optimization algorithm whose terms have no reasonable explanation. To find a new theory, theories need an explanation of the thinking process. The Feynman path integrals and the associated probability functionals may not make sense but the rules are fairly clear. "Just go for it" doesn't lead to a new and more reasonable theory.
Quantum mechanics does have a different problem which is that it needed some serious development in math; Feynman didn't like renormalization, and he was right about that. Better math foundations have to be worked out. Calculus existed for many, many decades as the calculation mechanism for physics problems before it was put on a rigorous mathematical foundation, though, so it's not the first time this has happened.
I think that a large part of what underlies the disinterest in the "what is it really" question in quantum mechanics is that we don't really have any idea what a "correct" - or even a good - answer would be.
The math already describes the data, so any interpretation of the math cannot be judged "correct" or "incorrect"on the basis of matching the data. And we already know that quantum mechanics violates our macro-scale intuitions, so we cannot judge "better" based on alignment with those intuitions.
I believe that Quantum Mechanics(QM) is a dead end and stops at the late Steven Weinberg's, Lectures On Quantum Mechanics. Prof. Weinberg also thought that a new theory is needed. QM and relativity may well be just algorithms analogous to the pre-Copernican planetary models.
So, algebra's essential for all math. But so's formal logic, which is missing from high school curricula.
I think inductive reasoning, standards of evidence, and critical thinking should be a mandatory part of the curriculum. Then logic. Also rhetoric, so people can defend themselves against fallacies. All are missing from the K-12 curriculum, or not taken seriously.
Then we can get into arithmetic and algebra. *Everyone* needs algebra. From there, the next step is probability.
And after that, statistics. Which is hard. And it's not really mathematics. It's a different, though closely related, field. Calculus doesn't tell you shit about statistics, though it can be one of many tools for trying to analyze statistical distributions. Half the statistics course needs to be on data collection, uncertainty, survey design, data presentation, all that stuff, none of which is actually math. I believe at this point that Bayesian approaches are better than frequentist, but both should be taught.
These need to be mandatory courses, and if the K-12s won't do it, they should be required college courses.
Geometry, trigonometry, and analytic geometry are important for those continuing in the sciences, though frankly not nearly as important as probability and statistics for everyday life. And analytic geometry isn't in the high school curriculum. Calculus... should come after analytic geometry, which it doesn't; when it comes after analytic geometry, it's easy.
Sigh. I was trying to restrain myself from commenting on the high school curriculum issue, but my will is broken. I agree with Peter Woit here:
"Many mathematicians and physicists have signed an Open Letter on K-12 Mathematics pointing to problems with attempts to reform mathematics education such as the California Mathematics Framework. For more about this, see the blog entry posted here and on Scott Aaronson’s blog, and more detail here.
While I’ve always had some sympathy for the general idea that there’s much that could be changed and improved about the US K-12 math curriculum, there’s a huge problem with all proposed changes based on the “algebra/pre-calculus/calculus sequence is too hard and not relevant to everyday life” argument. Students leaving high school without algebra and some pre-calculus are put in a position such that they’re unequipped to study calculus, and calculus is fundamental to learning physics. Without being able to learn physics, a huge range of possible fields of study and careers will be closed to them, from much of engineering through even going to medical school. Whatever change one makes to K-12 math education, it shouldn’t leave students entering college with a severely limited choice of fields they are prepared to study."
I took statistics and probability before I took calculus and didn't notice have problems later when I took calculus and a more advanced probability and statistics course. Understanding statistics at a useful level doesn't require calculus. It does help if it is grounded in an understanding of probability. There's nothing quite as convincing as watching one of those falling balls in a diagonal grid gizmos and seeing the bell curve appear again and again by pure chance. You can use statistics without even knowing the formula for the bell curve as long as you understand what the standard deviation is with a geometric intuition. Bayesian statistics are similarly useful.
There's a temptation for people who really understand mathematics to overthink things. My favorite was a tweet from a mathematician whose eight year old had asked "What is area?" John Cook took this and ran with it: “Well, first we have to define sigma algebras. They’re kinda like topologies, but closed under countable union and intersection instead of arbitrarily union and finite intersection. Anyway, a measure is a ...”
There's a mathematics joke that goes: "The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?" (attributed to Jerry Bona). The joke is that they're equivalent.
You can squeeze an axiom between the Axiom of Countable Choice and the Axiom of Choice, strictly stronger than the first and strictly weaker than the second: the Axiom of Dependent Choice.
We are talking high school, not university, (and even here calculus is not necessary for non-engineering or mathematics students).
In the real world, the problem with stats is not knowing formulae or how to derive them, it is applying the wrong tool in the toolbox. Almost everyone from scientists downwards seem to go for the easiest, off-the-shelf statistical tests and apply that. There was even a paper written many years ago about this problem in peer-reviewed journal papers. Very little thought, if any, is used to consider the underlying distribution. Worse, because statistical p values are important as a success measure, it is easy to use software that tests the data with a number of different tests to find one that "works". Data mining has made this worse as almost no one considers the needed corrections when applying lots of tests. A p-value of 0.05 means that random tests on a set of data might find a success 5% of the time.
I would rather have students be aware of distributions and be given a table of tests that are applicable for different types of distributions, and be prepared to use tests that are LEAST sensitive to those distributions.
Back when I was a student fighting cave bears, sample sizes were very small, and a non-parametric Chi-Square test was safe to use, even though we had to use tables to look up significance.
One other observation of university students I have taught. "Number blindness". Few students can even realize that a calculated value (due to a misused calculator) is "obviously" wrong. Many students just cannot grasp the relationships between numbers at all.
When I did the math at school with books of tables and a slide rule, the time needed to do the various operation was aided by some pre-thought about what the likely answer was. Using round numbers and extracting powers on paper was a necessary step. Reducing large numbers or many terms in a division by canceling common numerators and denominators simplified calculations to where some could be done with mental arithmetic alone. Today's students armed with fancy calculators are simply unaware of these simple tools to simplify calculations and ensure that the result is correct and not out by orders of magnitude.
Decades ago I read an article about kids not being able to make change for purchases as the checkout machines did the work. I also experienced high school dropouts unable to do simple geometry - e.g. how many rectangular tiles of dimension x,y will fit on a tray of dimension m,n. Forget these young adults doing statistics, even if a task depended on it.
The problem with quantum physics is that it is hard to find a good human scale analogy for understanding it. Every analogy one might choose has a comprehensible and visible mechanism underlying it. Try trains running every 15 minutes with a standard deviation of three minutes, and you can get a wave function for trains, but there are still physical trains out on the tracks with observable positions and velocities. In quantum mechanics, trains can only be observed when they arrive at the station, and no one is exactly sure what it means to observe or arrive or exactly what a station is.
"Shut up and compute" is useful advice if you want to use quantum physics, but it is philosophically unsatisfying. People have devised more satisfying philosophical approaches, but a lot depends on what one finds satisfying. As with most of philosophy, it's more about the human mind, and less about the world in which that mind exists. The Book of Job covers some of this. Some people feel it is important to understand the "essence" of things. Others don't. I'll stop here and quote Joan Didion, “What makes Iago evil? Some people ask. I never ask.”
(I'm not very religious, but now and then I pray to god or the gods that no one finds it necessary to do a version of Othello with a tedious backstory explaining Iago's motivation and, possibly worse, has Othello and Desdemona meeting cute.)
There are no good human-scale analogies for understanding it, because it is not a human-scale phenomenon. We have evolved—biologically and culturally—to understand human-scale things, and have no ability to analogize to understand this particular non-human-scale thing...
I tend to view it as a problem in art. We understand things through metaphors. This applies to both things of an ordinary human scale things as well as those more extreme. In the early 20th century, there was a whole artistic movement groping with understanding a new world of machines. Our avant garde is still wrestling with quantum mechanics. (i think stories of time travel, alternate worlds and contingency are in this genre.)
Basically, we are not smart enough to understand things other than through grand narratives and metaphors, often about journeying through physical space and grabbing and manipulating objects. Is the only solution to ditch civilization, go back up into the trees, and hope that we can evolve into smarter creatures?
Maybe I am misreading, but this seems a little too strong.
Yes, we have evolved to understand human-scale things, so there is no guarantee that we will be able intuitively to understand quantum physics. But it may turn out that at some point in the future we do come to understand it in a way that "makes sense" to us.
Or maybe not. But I (at least) am very skeptical of "we will never understand X" claims.
After all, the position of philosophy is to be permanently dissatisfied.
If so, it is highly unlikely to be in one-to-one correspondence with anything on the human scale...
This I agree with.
Perhaps it's my background as the child of a math professor and a statistics professor, but "shut up and compute" always seemed to be a satisfying description for Newtonian mechanics just as well as it is for Quantum mechanics.
Coefficients of friction? Eh, the models predict reality, so just go with it.
There is a very nice Feynman passage on this: how you cannot explain the more fundamental in terms of analogizing it to the less fundamental, precisely because the less fundamental than itself needs to be explained...
Indeed, quantum mechanics seems to be an algorithm for computation and does a good job on the Hydrogen atom. The Hydrogen anion H+ (ionized Hydrogen molecule ground state) seems to defeat it and requires a high-powered optimization algorithm whose terms have no reasonable explanation. To find a new theory, theories need an explanation of the thinking process. The Feynman path integrals and the associated probability functionals may not make sense but the rules are fairly clear. "Just go for it" doesn't lead to a new and more reasonable theory.
Quantum mechanics does have a different problem which is that it needed some serious development in math; Feynman didn't like renormalization, and he was right about that. Better math foundations have to be worked out. Calculus existed for many, many decades as the calculation mechanism for physics problems before it was put on a rigorous mathematical foundation, though, so it's not the first time this has happened.
I think that a large part of what underlies the disinterest in the "what is it really" question in quantum mechanics is that we don't really have any idea what a "correct" - or even a good - answer would be.
The math already describes the data, so any interpretation of the math cannot be judged "correct" or "incorrect"on the basis of matching the data. And we already know that quantum mechanics violates our macro-scale intuitions, so we cannot judge "better" based on alignment with those intuitions.
Which means that speculation is necessarily idle.
I believe that Quantum Mechanics(QM) is a dead end and stops at the late Steven Weinberg's, Lectures On Quantum Mechanics. Prof. Weinberg also thought that a new theory is needed. QM and relativity may well be just algorithms analogous to the pre-Copernican planetary models.
So, algebra's essential for all math. But so's formal logic, which is missing from high school curricula.
I think inductive reasoning, standards of evidence, and critical thinking should be a mandatory part of the curriculum. Then logic. Also rhetoric, so people can defend themselves against fallacies. All are missing from the K-12 curriculum, or not taken seriously.
Then we can get into arithmetic and algebra. *Everyone* needs algebra. From there, the next step is probability.
And after that, statistics. Which is hard. And it's not really mathematics. It's a different, though closely related, field. Calculus doesn't tell you shit about statistics, though it can be one of many tools for trying to analyze statistical distributions. Half the statistics course needs to be on data collection, uncertainty, survey design, data presentation, all that stuff, none of which is actually math. I believe at this point that Bayesian approaches are better than frequentist, but both should be taught.
These need to be mandatory courses, and if the K-12s won't do it, they should be required college courses.
Geometry, trigonometry, and analytic geometry are important for those continuing in the sciences, though frankly not nearly as important as probability and statistics for everyday life. And analytic geometry isn't in the high school curriculum. Calculus... should come after analytic geometry, which it doesn't; when it comes after analytic geometry, it's easy.
Sigh. I was trying to restrain myself from commenting on the high school curriculum issue, but my will is broken. I agree with Peter Woit here:
"Many mathematicians and physicists have signed an Open Letter on K-12 Mathematics pointing to problems with attempts to reform mathematics education such as the California Mathematics Framework. For more about this, see the blog entry posted here and on Scott Aaronson’s blog, and more detail here.
While I’ve always had some sympathy for the general idea that there’s much that could be changed and improved about the US K-12 math curriculum, there’s a huge problem with all proposed changes based on the “algebra/pre-calculus/calculus sequence is too hard and not relevant to everyday life” argument. Students leaving high school without algebra and some pre-calculus are put in a position such that they’re unequipped to study calculus, and calculus is fundamental to learning physics. Without being able to learn physics, a huge range of possible fields of study and careers will be closed to them, from much of engineering through even going to medical school. Whatever change one makes to K-12 math education, it shouldn’t leave students entering college with a severely limited choice of fields they are prepared to study."
Also, hiring Jo Boaler to redesign your math curriculum does look a lot like hiring Jordan Peterson to redesign your humanities curriculum: https://fillingthepail.substack.com/p/tessellated-with-good-intentions.
Touché…
I took statistics and probability before I took calculus and didn't notice have problems later when I took calculus and a more advanced probability and statistics course. Understanding statistics at a useful level doesn't require calculus. It does help if it is grounded in an understanding of probability. There's nothing quite as convincing as watching one of those falling balls in a diagonal grid gizmos and seeing the bell curve appear again and again by pure chance. You can use statistics without even knowing the formula for the bell curve as long as you understand what the standard deviation is with a geometric intuition. Bayesian statistics are similarly useful.
There's a temptation for people who really understand mathematics to overthink things. My favorite was a tweet from a mathematician whose eight year old had asked "What is area?" John Cook took this and ran with it: “Well, first we have to define sigma algebras. They’re kinda like topologies, but closed under countable union and intersection instead of arbitrarily union and finite intersection. Anyway, a measure is a ...”
https://twitter.com/JohnDCook/status/1322683274490269696
There's a mathematics joke that goes: "The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?" (attributed to Jerry Bona). The joke is that they're equivalent.
You can squeeze an axiom between the Axiom of Countable Choice and the Axiom of Choice, strictly stronger than the first and strictly weaker than the second: the Axiom of Dependent Choice.
https://en.wikipedia.org/wiki/Axiom_of_dependent_choice
I did not know that...
Re: Calculus and statistics
We are talking high school, not university, (and even here calculus is not necessary for non-engineering or mathematics students).
In the real world, the problem with stats is not knowing formulae or how to derive them, it is applying the wrong tool in the toolbox. Almost everyone from scientists downwards seem to go for the easiest, off-the-shelf statistical tests and apply that. There was even a paper written many years ago about this problem in peer-reviewed journal papers. Very little thought, if any, is used to consider the underlying distribution. Worse, because statistical p values are important as a success measure, it is easy to use software that tests the data with a number of different tests to find one that "works". Data mining has made this worse as almost no one considers the needed corrections when applying lots of tests. A p-value of 0.05 means that random tests on a set of data might find a success 5% of the time.
I would rather have students be aware of distributions and be given a table of tests that are applicable for different types of distributions, and be prepared to use tests that are LEAST sensitive to those distributions.
Back when I was a student fighting cave bears, sample sizes were very small, and a non-parametric Chi-Square test was safe to use, even though we had to use tables to look up significance.
One other observation of university students I have taught. "Number blindness". Few students can even realize that a calculated value (due to a misused calculator) is "obviously" wrong. Many students just cannot grasp the relationships between numbers at all.
When I did the math at school with books of tables and a slide rule, the time needed to do the various operation was aided by some pre-thought about what the likely answer was. Using round numbers and extracting powers on paper was a necessary step. Reducing large numbers or many terms in a division by canceling common numerators and denominators simplified calculations to where some could be done with mental arithmetic alone. Today's students armed with fancy calculators are simply unaware of these simple tools to simplify calculations and ensure that the result is correct and not out by orders of magnitude.
Decades ago I read an article about kids not being able to make change for purchases as the checkout machines did the work. I also experienced high school dropouts unable to do simple geometry - e.g. how many rectangular tiles of dimension x,y will fit on a tray of dimension m,n. Forget these young adults doing statistics, even if a task depended on it.
There is an excellent MIT course on "Street-Fighting Math", IIRC...
Agree with all observations by Alex Tolley
Drop calc, add stats, including the difficult non-math part of stats