Oh my God, he's taken me seriously. The guy's mad! 😂
See When Your Daughter Has Problems With Learning Times Tables. At 9 minutes 33 seconds, we can give people tools to look for and systematically explore the structure for these patterns. Here's just one example of something that could run on a smartphone, but what we need are tools which let them use novel schemes to illustrate patterns, and apply them in different places. For example, let them explore the patterns in repeating digits in the higher positions, as they change to different multiples and initial offsets.
So at 15 minutes 13 seconds, you might find your daughter comes to tell you something like this:
So you phone her mum and say "She's doing her times tables in base 324, and getting quite good at it!" And her mother says "but where does she get all the symbols from?" ... And you say, "I dunno, ask Harvey Friedman!" See A Concrete Way to See What Computable Functions Actually Are. And she says, "Harvey Friedman?! Are you letting her hang out with that CIA guy? ..." And you say "No, she just wrote a program which applied Cayley's Theorem to an arbitrary subgroup of Z and then searched the symmetries of that program for the first proof of the elements in Lagrange's Theorem, ... And her mother says, "Oh, that's OK. I was having nightmare visions of Sienna, ...." See The Weirdest Math Video You'll See Today.
And when it comes to algebra, we need to think in a similar way about how to teach algebra not as simply a mechanical method to automate calculation, but as a systematic method to automate proof. Start by having a look at how algebra is taught here: Charles L. Dodgson's Euclid Book V - Algebraically
... because then, work such as Jean-Yves Girard's Proof-theory and logical complexity will become much more widely understood!
The limitation to commensurable magnitudes is transcended in this text on Algebraic Geometry: An Elementary Treatise on Determinants: With Their Application to Simultaneous Linear Equations and Algebraic Geometry.
Now the problem is, .... somebody has banned such tools from being made available to the general population. Why? Because they are too powerful. See the following papers, from the bibliography to this essay I wrote in 2015: A Standard Basis for Standard Bases:
I think this suppression of these ideas was an exercise carried out at the request of, or certainly with assistance from, the Special Activities Division (SAD) of the CIA, later quite appropriately renamed "Shock and Awe". They are indeed shockingly and awesomely stupid people!
But there have been exceptions, such as Larry Paulson's 1981 Stanford PhD dissertation which was a very much watered-down version of the sorts of logical framework described by Cousot and Cousot in the above papers, and which is what presumably earned him the position of Research Assistant to Robin Milner on the Logic of Computable Functions project at Edinburgh in 1982. See Amy Zegart on Cyberwar. Another was the OTT tool, developed by Peter Sewell when he was having trouble getting another research grant after his hopeless project trying to determine the semantics of open source operating system TCP/IP stacks. IBM almost immediately awarded him a contract to determine the semantics of closed, proprietary CPU technology, ... then promptly sold the business, ...
See When Your Daughter Has Problems With Learning Times Tables. At 9 minutes 33 seconds, we can give people tools to look for and systematically explore the structure for these patterns. Here's just one example of something that could run on a smartphone, but what we need are tools which let them use novel schemes to illustrate patterns, and apply them in different places. For example, let them explore the patterns in repeating digits in the higher positions, as they change to different multiples and initial offsets.
So at 15 minutes 13 seconds, you might find your daughter comes to tell you something like this:
So you phone her mum and say "She's doing her times tables in base 324, and getting quite good at it!" And her mother says "but where does she get all the symbols from?" ... And you say, "I dunno, ask Harvey Friedman!" See A Concrete Way to See What Computable Functions Actually Are. And she says, "Harvey Friedman?! Are you letting her hang out with that CIA guy? ..." And you say "No, she just wrote a program which applied Cayley's Theorem to an arbitrary subgroup of Z and then searched the symmetries of that program for the first proof of the elements in Lagrange's Theorem, ... And her mother says, "Oh, that's OK. I was having nightmare visions of Sienna, ...." See The Weirdest Math Video You'll See Today.
And when it comes to algebra, we need to think in a similar way about how to teach algebra not as simply a mechanical method to automate calculation, but as a systematic method to automate proof. Start by having a look at how algebra is taught here: Charles L. Dodgson's Euclid Book V - Algebraically
... because then, work such as Jean-Yves Girard's Proof-theory and logical complexity will become much more widely understood!
The limitation to commensurable magnitudes is transcended in this text on Algebraic Geometry: An Elementary Treatise on Determinants: With Their Application to Simultaneous Linear Equations and Algebraic Geometry.
I think this suppression of these ideas was an exercise carried out at the request of, or certainly with assistance from, the Special Activities Division (SAD) of the CIA, later quite appropriately renamed "Shock and Awe". They are indeed shockingly and awesomely stupid people!
But there have been exceptions, such as Larry Paulson's 1981 Stanford PhD dissertation which was a very much watered-down version of the sorts of logical framework described by Cousot and Cousot in the above papers, and which is what presumably earned him the position of Research Assistant to Robin Milner on the Logic of Computable Functions project at Edinburgh in 1982. See Amy Zegart on Cyberwar. Another was the OTT tool, developed by Peter Sewell when he was having trouble getting another research grant after his hopeless project trying to determine the semantics of open source operating system TCP/IP stacks. IBM almost immediately awarded him a contract to determine the semantics of closed, proprietary CPU technology, ... then promptly sold the business, ...
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