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Cuchulainn
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Re: Philosophy of Mathematics

October 18th, 2020, 10:12 pm

'Good' intension (clear bondary, so Frege works)
// Point.hpp
//
// Generic point class. The first coordinate is of one
// type and the second coordinate is of the second type.
//
// Version 1: Distance() function contains a std function as
// argument
//
// (C) Datasim Education BV 2006-2013

#ifndef Point_HPP
#define Point_HPP

#include <iostream>
#include <functional>

template <class X=double> class Point
{
private:

	// The two coordinates
	X m_x;
	X m_y;

	// The embedded algorithm (strategy) object; each object gets its own copy
	std::function<double (const Point<X>& p1, const Point<X>& p2)> algo;

public:

	typedef std::function<double (const Point<X>& p1, const Point<X>& p2)> FunctionType;

	// Constructors & destructor
	Point(const FunctionType& algorithm);										// Default constructor
	Point(const X& first, const X& second, const FunctionType& algorithm);					// Constructor with coordinates
	Point(const Point<X>& source);					// Copy constructor
	virtual ~Point();								// Destructor

	// Selectors
	const X& First() const;								// Get first coordinates
	const X& Second() const;							// Get second coordinate

	// Modifiers
	void First(const X& val);						// Set first coordinate
	void Second(const X& val);						// Set second coordinates

	// Functions
	double Distance(const Point<X>& p2) const;		// Calculate distance

	// Assignment operator
	Point<X>& operator = (const Point<X>& source);

	template <class X> 
		friend std::ostream& operator << (std::ostream& os, const Point<X>& p);
};

#include "Point.cpp"

#endif

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Cuchulainn
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Re: Philosophy of Mathematics

October 18th, 2020, 10:15 pm

Good extension
Point<double> p1(1.0, 1.0, ExactDistance);
 Point<double> p2(p1); 

 std::cout << "p1: " << p1 << std::endl;
 std::cout << "p2: " << p2 << std::endl;
"Compatibility means deliberately repeating other people's mistakes."
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complyorexplain
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Re: Philosophy of Mathematics

October 19th, 2020, 10:39 am

Note it's not just Frege. Predicate calculus is the basis of all mathematical logic, including Goedel.

Propositional functions can include tense. 'is-a-philosopher(Trump) is in the present case, and all such functions are in the present tense unless specified otherwise, in which case you have tense logic.

https://plato.stanford.edu/entries/logic-temporal/

Which is also a fairly wide subject.
 
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Cuchulainn
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Re: Philosophy of Mathematics

October 19th, 2020, 1:19 pm

Note it's not just Frege. Predicate calculus is the basis of all mathematical logic, including Goedel.

Propositional functions can include tense. 'is-a-philosopher(Trump) is in the present case, and all such functions are in the present tense unless specified otherwise, in which case you have tense logic.

https://plato.stanford.edu/entries/logic-temporal/

Which is also a fairly wide subject.
Very interesting post.
Imagine imparting this to an OO programmer. Most OO languages don't support this. Once a president, always a president.
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Cuchulainn
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Re: Philosophy of Mathematics

October 19th, 2020, 7:36 pm

Could Frege concoct this?

Image
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Re: Philosophy of Mathematics

October 20th, 2020, 10:11 am

The ability to model time-dependent concepts is useful but the fatal flaw IMHO in Frege's Defining Attribute View is that concepts are not static; the are volatile and context-sensitive. Concepts are unstable.
The Explanation-based View remedies some of these shortcomings.
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complyorexplain
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Re: Philosophy of Mathematics

October 20th, 2020, 11:21 am

Look at §46 of *The Foundations of Arithmetic* where he considers, and resolves, the objection that the concept "inhabitant of Germany" changes its extension every year. 

He says that the extension of the concept "inhabitant of Germany at New Year 1883, Berlin time" is the same for all eternity.

*The Foundations of Arithmetic* is well worth reading as an introduction to his work.
 
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Re: Philosophy of Mathematics

October 20th, 2020, 2:16 pm

Reminded me of Lewis Carroll's classic:

Barry, Cole, and Dix, agreed, with two other friends of theirs, Lang and Mill, that the five should meet, every day, at a certain table d’hôte. Remembering how much amusement they had derived from their code of rules for walking-parties, they devised the following rules to be observed whenever beef appeared on the table:—

(1) If Barry takes salt, then either Cole or Lang takes one only of the two condiments, salt and mustard: if he takes mustard, then either Dix takes neither condiment, or Mill takes both.
(2) If Cole takes salt, then either Barry takes only one condiment, or Mill takes neither: if he takes mustard, then either Dix or Lang takes both.
(3) If Dix takes salt, then either Barry takes neither condiment or Cole take both: if he takes mustard, then either Lang or Mill takes neither.
(4) If Lang takes salt, then Barry or Dix takes only one condiment: if he takes mustard, then either Cole or Mill takes neither.
(5) If Mill takes salt, then either Barry or Lang takes both condiments: if he takes mustard, then either Cole or Dix takes only one.
The Problem is to discover whether these rules are compatible; and, if so, what arrangements are possible.

What's the answer? :-)

BTW, Frege's basic logic doesn't seem to apply in most real-world settings, even assuming that it can cover the temporal and context-depedent (whatever it means precisely) aspects. Think of all sorts of logical fallacies (conjunction, disjunction, anecdotal, ...) or even simple uncertainty (e.g. due to incomplete information).
One approach to these kinds of problems is the quantum probability picture (responses are vectors in the Hilbert space) and another is reinforcement learning.
Invoking the discussion on traders in the market, a fun and relatively simple method to model them is the multi-agent reinforcement learning - a modern alternative to old-school agent-based models.
 
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Re: Philosophy of Mathematics

October 21st, 2020, 8:54 am


He says that the extension of the concept "inhabitant of Germany at New Year 1883, Berlin time" is the same for all eternity.

*The Foundations of Arithmetic* is well worth reading as an introduction to his work.
Is that Eastern Standard Time, or Rocky Mountain Time?
I am not concerned with Frege's internal logic but the very fact that his concepts are unstable, e.g. can an inhabitant be a tax exile and lives in Germany for 121 days per year?

I don't have that book. But only Begriffschift. For me, Dedekind made Arithmetic respecatble.
Last edited by Cuchulainn on October 21st, 2020, 9:12 am, edited 3 times in total.
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Re: Philosophy of Mathematics

October 21st, 2020, 8:57 am

Frege laid the foundations of Functional Programming

Wie nun Funktionen von Gegenständen grundverschieden sind, so sind auch Funktionen, deren Argumente Funktionen sind und sein müssen, grundverschieden von Funktionen, deren Argumente Gegenstände sind und nichts anderes sein können. Diese nenne ich Funktionen erster, jene Funktionen zweiter Stufe.
G. Frege, “Funktion und Begriff”, 1891

aka higher-order functions. Brilliant.

Alonzo Church extended Frege's theory.

https://en.wikipedia.org/wiki/Frege%E2% ... _confusion
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Re: Philosophy of Mathematics

October 22nd, 2020, 1:46 pm

Can the existence of a mathematical entity be proved without defining it ?

Jacques Hadamard
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complyorexplain
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Re: Philosophy of Mathematics

October 22nd, 2020, 3:42 pm

Can the existence of a mathematical entity be proved without defining it ?

Jacques Hadamard
So did the square root of 2 exist before anyone defined it? I imagine so, at least if you buy into the Platonistic idea of mathematical entities.
 
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Re: Philosophy of Mathematics

October 22nd, 2020, 4:27 pm

Can the existence of a mathematical entity be proved without defining it ?

Jacques Hadamard
So did the square root of 2 exist before anyone defined it? I imagine so, at least if you buy into the Platonistic idea of mathematical entities.
I suspect the Babylonians discovered it when they built those ziggurats. In contrast to Plato, they had algorithms.

Hadamard was able to infer qualitative properties of PDEs w/o the need to construct a solution.
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Re: Philosophy of Mathematics

October 22nd, 2020, 4:37 pm

the world is discrete, so sqrt 2 cannot exist physically ? it is a probability = in the head of mathematicians, and a very good approximation in the macroscopic world !!

Weyl tile
 
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Re: Philosophy of Mathematics

October 22nd, 2020, 7:35 pm

the world is discrete, so sqrt 2 cannot exist physically ? it is a probability = in the head of mathematicians, and a very good approximation in the macroscopic world !!

Weyl tile
I's sure with a bit of training we could get them into piysicists' crania as well.
"Compatibility means deliberately repeating other people's mistakes."
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