Indiana Jade on Information and Representation
This is really interesting!
More interesting than Neo-whatsits, anyway, ...
Here's an amateur doing the same scene:
See Rewrite - SciFi Short for more on presentation and representation.
See Greek numerals for an authoritative description of how the Greeks represented numbers. But I wouldn't have done it like that if I were Greek. I would have cooked up a rule to save myself from having to invent new names for bigger numbers which are obviously fractal. See
... and:
... and:
... and on Hausdorff dimension see Nathan Has 1,000 Subscribers:
And on variable bases, see Asymmetric numeral systems which are form of entropy encoding. On their uses in probability, see Measuring Ignorance and look up Kullback–Leibler divergence; then see Aristotle on The Continuum:
At 8 minutes 20 seconds (500 seconds) Norman says "Eventually the numbers I'm going to get are going to be too big to fit on my hard drive." How does he know that there is no compression scheme which will allow him to store any of those numbers in a finite amount of space? Because most of them (apart from a set of measure zero) are uncomputable?
Zeilberger makes the dual of the same mistake Wildberger makes. In Zeilberger's objection to the existence of uncomputable numbers, he assumes the greatest integer is a constant, whereas Wildberger assumes that the amount of data compression available to store numbers on his hard drive is constant. The reality however is that Zeilberger's constant Z is a function of the amount of information in the interpretive context Γ of the system. See On Free Will and Conscious Awareness and On Tarski's Semantic Definition of Truth 'Convention-T'.
Now think about SPACE complexity while you watch this:
For more on constructing a complete lattice using a Galois Connection to construct a fixpoint operator, see Edsger W. Dijkstra on Reasoning about Processes then watch:
Then look for a way to define turing machines as regular expressions with recursive back-references and substitution as described in this post: Logic. You could, for example, represent the current head pisition by a special sequence >_< say, where _ is any symbol from the alphabet.
Now, to understand how formal cause works, and why mathematics is real, all you need to do is listen to Hilary Putnam explaining the idea of functional isomorphism in semantic models. This talk is introduced by the philosopher John Searle, I think.
On the "sloppy, but not fatally sloppy" argument 11 minutes 42 seconds, that we need not know exactly what it is that the functional isomorphism is an isomorphism of: this was resolved by Robin Milner who introduced the notion of polymorphic type schemes to describe a greatest fixpoint where the foundational semantics are specified by type variables in the object language. These abstract structures, instead of being equivalent "up to isomorphism", could be shown to be so "down to polymorphism". See this description of Hindley-Milner type inference and Principal Type Schemes for Functional Programs by Damas and Milner.
This is more or less what the programming language Standard ML implements as abstract types. See The Definition of Standard ML by Robin Milner, Mads Tofte, Robert Harper and David MacQueen and An ideal model for recursive polymorphic types by MacQueen, Plotkin and Sethi. This is a significant part of what Girard et al explore in the context of logical complexity in Girard, Lafont & Taylor's "Proofs and Types", abstract types on the other side of the Curry-Howard Polymorphism turn out to be closely related to Gödel's Incompleteness and Consistency Theorems. See this short note written in 2014. I guess this is why Jade is holding up one finger in front of a sort of Y-combinator and looking deadly serious on the cover graphic of the video above. On the final page of Proofs and Representations there is a 42 line formal proof of a fixpoint combinator type derived automatically by the Hindley-Milner-Robinson type inference algorithm.
The reality of functional isomorphisms then comes directly from the reality of conscious experience as mediated by language. This is very clearly explained by Searle here: John Searle on Philosophy of Language.
See Proofs and Representations for more on computability and a very general form of functional isomorphism called bisimulation.
The connection with information and physical reality is then fairly straightforward. See Julian Barbour Making Sense of Information. For the physical perspective, which basically comes down to the necessity that any measurement apparatus, whether quantum or classical, has a well-defined semantic meaning, is explored here: Relational Semantics subtitled "The Evils of Lego".
To see how this all ties up with Ada Byron's analytic functional calculus, watch this:
Then see The Right Way to Formalise Mathematics.
For some ideas about computer architecture optimised for such applications, see Ben Eater on Running a Breadboard at 1MHz and for some application areas of this, see Tim Palmer - Greenhouse Blues and Sabine Hossenfelder - Lost in Math.
There's a lot to be said for the Romany Gypsy education system:
This 'smoking gun' post, brought to you by:
and
As to Zeilberger's assertion that 2+2=4, ... What if Z=3?
See Trinity (Andrei Rublev).
See Hopi Prophecies and the Fifth World of the Australian Aboriginals.
More interesting than Neo-whatsits, anyway, ...
Here's an amateur doing the same scene:
See Rewrite - SciFi Short for more on presentation and representation.
... and:
... and:
... and on Hausdorff dimension see Nathan Has 1,000 Subscribers:
And on variable bases, see Asymmetric numeral systems which are form of entropy encoding. On their uses in probability, see Measuring Ignorance and look up Kullback–Leibler divergence; then see Aristotle on The Continuum:
At 8 minutes 20 seconds (500 seconds) Norman says "Eventually the numbers I'm going to get are going to be too big to fit on my hard drive." How does he know that there is no compression scheme which will allow him to store any of those numbers in a finite amount of space? Because most of them (apart from a set of measure zero) are uncomputable?
Zeilberger makes the dual of the same mistake Wildberger makes. In Zeilberger's objection to the existence of uncomputable numbers, he assumes the greatest integer is a constant, whereas Wildberger assumes that the amount of data compression available to store numbers on his hard drive is constant. The reality however is that Zeilberger's constant Z is a function of the amount of information in the interpretive context Γ of the system. See On Free Will and Conscious Awareness and On Tarski's Semantic Definition of Truth 'Convention-T'.
Now think about SPACE complexity while you watch this:
For more on constructing a complete lattice using a Galois Connection to construct a fixpoint operator, see Edsger W. Dijkstra on Reasoning about Processes then watch:
Then look for a way to define turing machines as regular expressions with recursive back-references and substitution as described in this post: Logic. You could, for example, represent the current head pisition by a special sequence >_< say, where _ is any symbol from the alphabet.
Now, to understand how formal cause works, and why mathematics is real, all you need to do is listen to Hilary Putnam explaining the idea of functional isomorphism in semantic models. This talk is introduced by the philosopher John Searle, I think.
On the "sloppy, but not fatally sloppy" argument 11 minutes 42 seconds, that we need not know exactly what it is that the functional isomorphism is an isomorphism of: this was resolved by Robin Milner who introduced the notion of polymorphic type schemes to describe a greatest fixpoint where the foundational semantics are specified by type variables in the object language. These abstract structures, instead of being equivalent "up to isomorphism", could be shown to be so "down to polymorphism". See this description of Hindley-Milner type inference and Principal Type Schemes for Functional Programs by Damas and Milner.
This is more or less what the programming language Standard ML implements as abstract types. See The Definition of Standard ML by Robin Milner, Mads Tofte, Robert Harper and David MacQueen and An ideal model for recursive polymorphic types by MacQueen, Plotkin and Sethi. This is a significant part of what Girard et al explore in the context of logical complexity in Girard, Lafont & Taylor's "Proofs and Types", abstract types on the other side of the Curry-Howard Polymorphism turn out to be closely related to Gödel's Incompleteness and Consistency Theorems. See this short note written in 2014. I guess this is why Jade is holding up one finger in front of a sort of Y-combinator and looking deadly serious on the cover graphic of the video above. On the final page of Proofs and Representations there is a 42 line formal proof of a fixpoint combinator type derived automatically by the Hindley-Milner-Robinson type inference algorithm.
The reality of functional isomorphisms then comes directly from the reality of conscious experience as mediated by language. This is very clearly explained by Searle here: John Searle on Philosophy of Language.
See Proofs and Representations for more on computability and a very general form of functional isomorphism called bisimulation.
The connection with information and physical reality is then fairly straightforward. See Julian Barbour Making Sense of Information. For the physical perspective, which basically comes down to the necessity that any measurement apparatus, whether quantum or classical, has a well-defined semantic meaning, is explored here: Relational Semantics subtitled "The Evils of Lego".
To see how this all ties up with Ada Byron's analytic functional calculus, watch this:
Then see The Right Way to Formalise Mathematics.
For some ideas about computer architecture optimised for such applications, see Ben Eater on Running a Breadboard at 1MHz and for some application areas of this, see Tim Palmer - Greenhouse Blues and Sabine Hossenfelder - Lost in Math.
There's a lot to be said for the Romany Gypsy education system:
This 'smoking gun' post, brought to you by:
As to Zeilberger's assertion that 2+2=4, ... What if Z=3?
See Trinity (Andrei Rublev).
See Hopi Prophecies and the Fifth World of the Australian Aboriginals.
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