Foundations of Mathematics.
Foundations of Mathematics.
What are the foundations of mathematics? Does mathematics require foundations?
I hold that recourse to logic, sets or Peano's axioms are all spurious and unnecessary.
What is significant about mathematics is the relationship between the symbols. The symbols and the relationships between them and the processes by which theorems can be generated can all be specified by axioms that are presented without foundation or even validation.
What then counts is the ability of such an axiomatic system, using its internal logic, to generate theorems.
These theorems can then be used as a basis for mapping the abstract maths onto concepts of the real world. In this way mathematics can be used for such simple things as counting apples and dividing them evenly among numbers of people and also for more complex things such as the magnetic field around an inductor in an electrical circuit.
It is this mapping process that provides all the justification that mathematics needs.
Any comments?
I hold that recourse to logic, sets or Peano's axioms are all spurious and unnecessary.
What is significant about mathematics is the relationship between the symbols. The symbols and the relationships between them and the processes by which theorems can be generated can all be specified by axioms that are presented without foundation or even validation.
What then counts is the ability of such an axiomatic system, using its internal logic, to generate theorems.
These theorems can then be used as a basis for mapping the abstract maths onto concepts of the real world. In this way mathematics can be used for such simple things as counting apples and dividing them evenly among numbers of people and also for more complex things such as the magnetic field around an inductor in an electrical circuit.
It is this mapping process that provides all the justification that mathematics needs.
Any comments?

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Re: Foundations of Mathematics.
Can there be a foundation that doesn't include infinity in it
or can a foundation include infinity that doesn't lead to paradoxes? Does a foundation exist that's totally consistent? Do all math problems have explicit solutions?
Can currently difficult theorems be simplified? (some mathematicians think the ABC conjecture can help). Can a math system large enough to include arithmetic need to refer to a larger system for all of its proofs (Godel's theorem).
What about Langland's program? It's conjectured that different branches of math can be translated into one another (such as modular forms and elliptic equations). The importance and advantage is where a problem that may be insoluble in one branch can be solved in another and then translated back to the first branch; for those familiar with Laplace and reverse Laplace transforms, this is a minor example).
How many types of multigrade equations are there? Is it infinite? (an area I've been researching, for anyone interested I'll put them up here as I've discovered some interesting properties regarding them).
PhilX
or can a foundation include infinity that doesn't lead to paradoxes? Does a foundation exist that's totally consistent? Do all math problems have explicit solutions?
Can currently difficult theorems be simplified? (some mathematicians think the ABC conjecture can help). Can a math system large enough to include arithmetic need to refer to a larger system for all of its proofs (Godel's theorem).
What about Langland's program? It's conjectured that different branches of math can be translated into one another (such as modular forms and elliptic equations). The importance and advantage is where a problem that may be insoluble in one branch can be solved in another and then translated back to the first branch; for those familiar with Laplace and reverse Laplace transforms, this is a minor example).
How many types of multigrade equations are there? Is it infinite? (an area I've been researching, for anyone interested I'll put them up here as I've discovered some interesting properties regarding them).
PhilX

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 Location: Australia
Re: Foundations of Mathematics.
I agree with every word. Mathematics is a mapping tool and thus intrinsically tautologous. Such a tool can only be used to map a narrative of the world which must first be specified by the observer of it.A_Seagull wrote:What are the foundations of mathematics? Does mathematics require foundations?
I hold that recourse to logic, sets or Peano's axioms are all spurious and unnecessary.
What is significant about mathematics is the relationship between the symbols. The symbols and the relationships between them and the processes by which theorems can be generated can all be specified by axioms that are presented without foundation or even validation.
What then counts is the ability of such an axiomatic system, using its internal logic, to generate theorems.
These theorems can then be used as a basis for mapping the abstract maths onto concepts of the real world. In this way mathematics can be used for such simple things as counting apples and dividing them evenly among numbers of people and also for more complex things such as the magnetic field around an inductor in an electrical circuit.
It is this mapping process that provides all the justification that mathematics needs.
Any comments?
Re: Foundations of Mathematics.
Math is also a "construction" tool. Without an a preexisting blueprint which is all math, the bridge, building or whatever could not actualize into a 3 dimensional structure. Math and modelling are the preeminent disciplines in the reverse engineering of nature through observation and the description of every structure  a word whose meaning is implicitly mathematical  in describing the known architectures of nature and as the starting point in creating our own from microprocessors to buildings and bridges.
The constantly reiterated quote of Einstein the god, "with math you can prove anything" requires examination and not just simple minded acceptance or as a form of denigration. With math you may indeed prove the model in many cases but that does not mean that the model itself is correct based on what it meant to describe...though it may be good for another movie.
Without math and modelling nothing is possible not even chronology. The moment someone had to count something math became indispensable piling on layers of sophistication through the centuries and here we are ready, as Greta likes to point out, to deflect the most dangerous of NEO's.
The constantly reiterated quote of Einstein the god, "with math you can prove anything" requires examination and not just simple minded acceptance or as a form of denigration. With math you may indeed prove the model in many cases but that does not mean that the model itself is correct based on what it meant to describe...though it may be good for another movie.
Without math and modelling nothing is possible not even chronology. The moment someone had to count something math became indispensable piling on layers of sophistication through the centuries and here we are ready, as Greta likes to point out, to deflect the most dangerous of NEO's.
Re: Foundations of Mathematics.
I'm sorry but I don't get your point at all. Does it have anything to do with the foundations of mathematics?Dubious wrote:Math is also a "construction" tool. Without an a preexisting blueprint which is all math, the bridge, building or whatever could not actualize into a 3 dimensional structure. Math and modelling are the preeminent disciplines in the reverse engineering of nature through observation and the description of every structure  a word whose meaning is implicitly mathematical  in describing the known architectures of nature and as the starting point in creating our own from microprocessors to buildings and bridges.
The constantly reiterated quote of Einstein the god, "with math you can prove anything" requires examination and not just simple minded acceptance or as a form of denigration. With math you may indeed prove the model in many cases but that does not mean that the model itself is correct based on what it meant to describe...though it may be good for another movie.
Without math and modelling nothing is possible not even chronology. The moment someone had to count something math became indispensable piling on layers of sophistication through the centuries and here we are ready, as Greta likes to point out, to deflect the most dangerous of NEO's.
Re: Foundations of Mathematics.
I was admittedly too indirect in my response to your OP. I usually research history to discover the foundations of anything.
What you describe, as I see it, are already advanced notions, a consequence of that which constitutes the foundation. Since even advanced animals have a sense of number perhaps the foundational context for humans should be the equation format beginning with the most paradigmatic of ALL the ones to follow: 1 + 1 = 2.
It's what Bertrand Russell, over half way into the first volume of his Principia Mathematica, choice to prove by pure logic long after its established acceptance. Being the most precursory equation it doesn't simply count two entities summarized in one symbol but most importantly the relationship of equivalence in its most primal form. It's of a Class which all future equations inherit.
What you describe, as I see it, are already advanced notions, a consequence of that which constitutes the foundation. Since even advanced animals have a sense of number perhaps the foundational context for humans should be the equation format beginning with the most paradigmatic of ALL the ones to follow: 1 + 1 = 2.
It's what Bertrand Russell, over half way into the first volume of his Principia Mathematica, choice to prove by pure logic long after its established acceptance. Being the most precursory equation it doesn't simply count two entities summarized in one symbol but most importantly the relationship of equivalence in its most primal form. It's of a Class which all future equations inherit.
Re: Foundations of Mathematics.
Yes absolutely. This is what I was referring to when I said that the basis of mathematics was the relationship between symbols.Dubious wrote:I perhaps the foundational context for humans should be the equation format beginning with the most paradigmatic of ALL the ones to follow: 1 + 1 = 2.
.
In effect "1+1=2" would be one of the axioms of the system.
 Arising_uk
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Re: Foundations of Mathematics.
How did all the 'natives' build their bridges and huts then or did I miss your point?Dubious wrote:... Without an a preexisting blueprint which is all math, the bridge, building or whatever could not actualize into a 3 dimensional structure. ...
Re: Foundations of Mathematics.
Which mathematic?A_Seagull wrote:What are the foundations of mathematics? Does mathematics require foundations?
I hold that recourse to logic, sets or Peano's axioms are all spurious and unnecessary.
What is significant about mathematics is the relationship between the symbols. The symbols and the relationships between them and the processes by which theorems can be generated can all be specified by axioms that are presented without foundation or even validation.
What then counts is the ability of such an axiomatic system, using its internal logic, to generate theorems.
These theorems can then be used as a basis for mapping the abstract maths onto concepts of the real world. In this way mathematics can be used for such simple things as counting apples and dividing them evenly among numbers of people and also for more complex things such as the magnetic field around an inductor in an electrical circuit.
It is this mapping process that provides all the justification that mathematics needs.
Any comments?
As you correctly introduced your title, mathematics are actually plural, what is a sweet adjective to mean that they are not unified. So nothing good comes at this point.
And they will NEVER be unified; the Professor who waited for their unification did not even understood his own groundwork.
Analysis is  at best!  a simplification of a given equation;
Algebra  in accordance with Geometry  work in the opposite sense: they state some "foundations", on which they construct. This way, they are in the impossibility to show that their ground is false, they only can give the impression that they "confirm" this ground.
The story of these last domains are as of a little girl (because in french, geometry is female;)  named Geometry  which is very capricious. She gets what she needs, but she shows only what she want.
In the way mainstream logic are (see: viewtopic.php?f=26&t=19705#p276215) "false yields to true" & "true cannot yield to false", only mean that these last "mathematical" domains can establish anything they want. But, also, once they feel well on a way, they will never change the way.
And this is dramatical. That means that the "demonstration" is a pseudonotion. Once you think you are true, you are in the total impossibility to show what is false.
This is the way of nowadays science.
Moreover, it happens everyday, that "scientists" accord much credit to a theory, because by a mean or another, the theory is rendered as "looking beautiful".
The analysis is the sole mathematic, because in its best way, it does NOT INVENT INFORMATION.
Re: Foundations of Mathematics.
it seems that mathematics is founded on even absolutes of division multiplication addition and subtraction.
Maybe the universe contains partialities and more than wholes of division subtraction addition and multiplication. Call it fragnitudes and magnitudes.
Maybe the universe contains partialities and more than wholes of division subtraction addition and multiplication. Call it fragnitudes and magnitudes.
Re: Foundations of Mathematics.
maybe the universe acts in divisomultiples, divisoadditions combinations and sequence formulas and such. Maybe the universe acts in a different order than the order of operations. Like a funkitified illogic.
Last edited by osgart on Wed Oct 19, 2016 11:29 pm, edited 1 time in total.
Re: Foundations of Mathematics.
maybe we should use the absolute logic of math and compare it to a science of the illogical.
Re: Foundations of Mathematics.
maybe there are new logic frontiers in comparing the logical to the illogical. Abstract worlds trying to stare you in the face.
Re: Foundations of Mathematics.
mind blowing differentialities in the world of the illogical compared to the logical. Take the absolute and throw it off a bit and see what would happen.
Re: Foundations of Mathematics.
osgart wrote:maybe the universe acts in divisomultiples, divisoadditions combinations and sequence formulas and such for each single operation symbol. Maybe the universe acts in a different order than the order of operations. Like a funkitified illogic.
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