5 ms·

> I can not accept a claim that mathematics is discovered Why? Finding out that Pi is irrational and then transcendental are discoveries. I won't be able to m

by mnl 6y ago

> I can not accept a claim that mathematics is discovered Why? Finding out that Pi is irrational and then transcendental are discoveries. I won't be able to make anything work if I fork mathematics to make it rational. Same thing that finding out that if you tie a string to a stake and then go around keeping it tight, you have to walk a distance that is two times pi times the length of the string to get where you started. I don't need a language for that, just more pieces of string.

throwaway17_17 6y ago
Pi is a concept that is divorced from reality, Pi is irrational because a chain of logical inferences and applications result in the definition. The definition of Pi is predicated on a geometric description of a circle which is distinctly non-physical. The entire meaning of Pi is based on a logical chain of defined symbols, that while meaningful to some conscious thinker are not observations of physical reality. The geometric conception of a circle as an infinite collection of points, which are inherently not physically possible objects, exhibits a ratio between its radius and its diameter, which is called Pi, and that ratio needs to be expressed by a number which is definitionally irrational (the definition of which is predicated on a definition of Natural numbers (via a definition of the Real numbers)). These symbols are merely the language we use to describe logical relations between various symbols, that while possibly applicable, with prediction and accuracy preconditions, to physical reality, are merely a logic used to describe natural numbers and the properties thereof. It is this complete logical decoupling from any physical, observable reality, that I cannot accept mathematics as anything other than an, ideally, internally consistent, logic used to define and discuss symbolic logical objects. And I am of the opinion that logics are merely invented by conscious beings as means to systematize areas of discourse and to derive internally consistent inferences and implications built there on. I would agree that if irrational numbers were prohibited from being inferred in the logical system of math then some things would not work within the logic. Because a change in the axiomatic or inferential rules of a logic are changed the logic as a whole changes. But, if, as you describe it, math was forked to be limited to rationals, and the precision of those radicals was calibrated to be on the order of 1/10 of a Planck length, I am not convinced that any equation attempting to describe 3 dimensional space would be inaccurate. However, I could wrong.
- mensetmanusman 6y ago
  You lost me here: “ Pi is a concept that is divorced from reality”
  - ahelwer 6y ago
    Then you missed a pretty good post. You need at most ten decimal places on pi to get atomic-scale precision in anything to do with reality.
  - throwaway17_17 6y ago
    Pi, or the ratio between a geometrically defined circle’s circumference and radius ONLY holds as a truth in so much as one is discussing geometric circles. Pi does not describe the physical world, it describes a relationship between logical objects.
    - beagle3 6y ago
      That's true. But it is uncommon in my experience to actually define Pi as this ratio, if you build math axiomatically; Rather, the "shortest way" to get to pi which is well defined is by first defining the exponential function ( \exp x = \sum_i=0^\inf \frac{x^i}{\fact i} ) with all the pre-requisite for that (numbers with order, addition and multiplication; limits and convergence; then imaginary numbers). Then you define pi to be the smallest positive number such that exp(2pii) is 1, and e to be exp(1). All the properties of pi follow "easily", including it being the ratio of am euclidean circle's circumference to a diameter. The thing is, in math, all of these things end up the same regardless of where you start; Whether you start with a geometric definition of a circle and work hard to discover said ratio, or you start with exp(), you'll end up with pi=3.14159.... and it having the other properties. It is in that sense, not arbitrary. You could (and would) take a step back, and say that being euclidean is arbitrary - which is true; but any description compatible with euclidean axioms will get the same value of pi as the ratio of circumference to diameter; and that value will be the same of the exp() value that has no concept whatsoever of euclidean space. It is in that sense that math is "discovered" - there is no euclidean construction in which pi is different. There is no peano-compatible (set, games, p-adic, or other) construction of natural numbers in which primes do not exist. The peano axioms themselves are, indeed, logically arbitrary. But the "discovery" is that "peano -> existance of prime numbers" -- and that does not depend on language.
ahelwer 6y ago
You might be interested in an influential essay called "Higher-order truths about chmess" by Dan Dennett. Chmess is a hypothetical variant of chess with slightly altered rules. There are infinite interesting theorems you can prove about chmess. But what is the value of doing so? Are these theorems discoveries? Pi is not calculated by running around in a circle with a string (nor by any other real-world measurement), it's calculated by setting up a rule-based system (math) and following the rules to derive its value. You can get pretty far in the real world by using a rational value for pi - ten decimal places should suffice for atomic-level precision, I believe.
- mnl 6y ago
  The thing is Pi is ubiquitous. One of the interesting consequences of the recent redefinition of the SI units is that now h, k, e and a few others are considered exact constants, just like c. A caveat: physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself, whatever that might be it's something dealt with epoché. So now the Stefan-Boltzmann constant, which involves powers of c, k and h is exact. It also involves the fifth power of Pi (through the value of the Riemann zeta fuction for argument 4). As you can measure the Stefan-Boltzmann constant being creative with a radiometer, you could determine increasingly accurate values of Pi experimentally. In the sense of magnitudes as used in physics, which is a fruitful approach to being quantitative in this confusing physical world, this "thing" we call "Pi" behaves like a dimensionless physical constant. Before you'd object, if we came up with more precise theories, experience shows that that would mean corrections to the laws implied above. It also shows that when this happens instances of "Pi" don't deviate from the Platonic value of Pi.
  - dfischer 6y ago
    “ physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself” This is entirely oxymoronic and double thinking and exactly the type of mental gymnastics that manipulates what truth is. Physics is based on the physical because it was built upon observation of reality using our senses. Yet modern physics is based on the imaginative and has nothing to do with the physical. Any measured unit is imaginary. The only thing that exists is the measure. Exists is based on physical. To be is to exist. To exist must be within physical reality. Without this distinction then we might as well give into the esoteric notion that everything is entirely a mental construct and nothing actually “is.” How far we’ve come from the physical to the mental and then lack the nuance to distinguish as such and furthermore blur the lines is a disgrace to critical thinking and cognition.

by mnl 6y ago

throwaway17_17 6y ago

Pi is a concept that is divorced from reality, Pi is irrational because a chain of logical inferences and applications result in the definition. The definition of Pi is predicated on a geometric description of a circle which is distinctly non-physical. The entire meaning of Pi is based on a logical chain of defined symbols, that while meaningful to some conscious thinker are not observations of physical reality. The geometric conception of a circle as an infinite collection of points, which are inherently not physically possible objects, exhibits a ratio between its radius and its diameter, which is called Pi, and that ratio needs to be expressed by a number which is definitionally irrational (the definition of which is predicated on a definition of Natural numbers (via a definition of the Real numbers)). These symbols are merely the language we use to describe logical relations between various symbols, that while possibly applicable, with prediction and accuracy preconditions, to physical reality, are merely a logic used to describe natural numbers and the properties thereof. It is this complete logical decoupling from any physical, observable reality, that I cannot accept mathematics as anything other than an, ideally, internally consistent, logic used to define and discuss symbolic logical objects. And I am of the opinion that logics are merely invented by conscious beings as means to systematize areas of discourse and to derive internally consistent inferences and implications built there on. I would agree that if irrational numbers were prohibited from being inferred in the logical system of math then some things would not work within the logic. Because a change in the axiomatic or inferential rules of a logic are changed the logic as a whole changes. But, if, as you describe it, math was forked to be limited to rationals, and the precision of those radicals was calibrated to be on the order of 1/10 of a Planck length, I am not convinced that any equation attempting to describe 3 dimensional space would be inaccurate. However, I could wrong.

mensetmanusman 6y ago
You lost me here: “ Pi is a concept that is divorced from reality”
- ahelwer 6y ago
  Then you missed a pretty good post. You need at most ten decimal places on pi to get atomic-scale precision in anything to do with reality.
- throwaway17_17 6y ago
  Pi, or the ratio between a geometrically defined circle’s circumference and radius ONLY holds as a truth in so much as one is discussing geometric circles. Pi does not describe the physical world, it describes a relationship between logical objects.
  - beagle3 6y ago
    That's true. But it is uncommon in my experience to actually define Pi as this ratio, if you build math axiomatically; Rather, the "shortest way" to get to pi which is well defined is by first defining the exponential function ( \exp x = \sum_i=0^\inf \frac{x^i}{\fact i} ) with all the pre-requisite for that (numbers with order, addition and multiplication; limits and convergence; then imaginary numbers). Then you define pi to be the smallest positive number such that exp(2pii) is 1, and e to be exp(1). All the properties of pi follow "easily", including it being the ratio of am euclidean circle's circumference to a diameter. The thing is, in math, all of these things end up the same regardless of where you start; Whether you start with a geometric definition of a circle and work hard to discover said ratio, or you start with exp(), you'll end up with pi=3.14159.... and it having the other properties. It is in that sense, not arbitrary. You could (and would) take a step back, and say that being euclidean is arbitrary - which is true; but any description compatible with euclidean axioms will get the same value of pi as the ratio of circumference to diameter; and that value will be the same of the exp() value that has no concept whatsoever of euclidean space. It is in that sense that math is "discovered" - there is no euclidean construction in which pi is different. There is no peano-compatible (set, games, p-adic, or other) construction of natural numbers in which primes do not exist. The peano axioms themselves are, indeed, logically arbitrary. But the "discovery" is that "peano -> existance of prime numbers" -- and that does not depend on language.

ahelwer 6y ago

You might be interested in an influential essay called "Higher-order truths about chmess" by Dan Dennett. Chmess is a hypothetical variant of chess with slightly altered rules. There are infinite interesting theorems you can prove about chmess. But what is the value of doing so? Are these theorems discoveries? Pi is not calculated by running around in a circle with a string (nor by any other real-world measurement), it's calculated by setting up a rule-based system (math) and following the rules to derive its value. You can get pretty far in the real world by using a rational value for pi - ten decimal places should suffice for atomic-level precision, I believe.

mnl 6y ago
The thing is Pi is ubiquitous. One of the interesting consequences of the recent redefinition of the SI units is that now h, k, e and a few others are considered exact constants, just like c. A caveat: physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself, whatever that might be it's something dealt with epoché. So now the Stefan-Boltzmann constant, which involves powers of c, k and h is exact. It also involves the fifth power of Pi (through the value of the Riemann zeta fuction for argument 4). As you can measure the Stefan-Boltzmann constant being creative with a radiometer, you could determine increasingly accurate values of Pi experimentally. In the sense of magnitudes as used in physics, which is a fruitful approach to being quantitative in this confusing physical world, this "thing" we call "Pi" behaves like a dimensionless physical constant. Before you'd object, if we came up with more precise theories, experience shows that that would mean corrections to the laws implied above. It also shows that when this happens instances of "Pi" don't deviate from the Platonic value of Pi.
- dfischer 6y ago
  “ physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself” This is entirely oxymoronic and double thinking and exactly the type of mental gymnastics that manipulates what truth is. Physics is based on the physical because it was built upon observation of reality using our senses. Yet modern physics is based on the imaginative and has nothing to do with the physical. Any measured unit is imaginary. The only thing that exists is the measure. Exists is based on physical. To be is to exist. To exist must be within physical reality. Without this distinction then we might as well give into the esoteric notion that everything is entirely a mental construct and nothing actually “is.” How far we’ve come from the physical to the mental and then lack the nuance to distinguish as such and furthermore blur the lines is a disgrace to critical thinking and cognition.