Theists Equivocating the Empirical with the Transcendental

[godelian]

The best theory we have so far is the scientific Big Bang where the origin is "physical" as defined below.

The Big Bang model describes some of the early history of the universe but not its cause. Where is the reproducible experimental test report for the cause of the Big Bang? When was this experimental test report produced and when reproduced?

The scientific model [& mathematic] is the most reliable at present.

For most questions about the physical universe, the answer cannot be justified with a reproducible experimental test report.

If you don't agree what non-scientific model are you relying upon.

You do not have an experimental test report to justify anything you say about the subject, simply, because such experimental test report does not even exist.

I had pointed out what is mathematics [regardless of whatever the formalism] it is ultimately reducible to the empirical and the physical on a priori basis.

This view is wrong. Mathematics is ultimately reducible to systematic string manipulation, and nothing else. Furthermore, empiricism is forbidden in mathematics, just like it is forbidden in all branches of Pure Reason.

Note Kant's argument on how Science and Mathematics is possible on a synthetic a priori basis, while metaphysics [ontological] is not possible.

Incorrect about science.

Science is absolutely not divorced from sensory input and is therefore not Pure Reason at all. Hence, in Kantian lingo, science is not synthetic a priori.

Regardless of whatever the mathematics, it is reducible to the empirical a priori.

You are in denial of David Hilbert's formalist conclusions on the matter. You are in denial of one of the core ontological principles in mathematics: empiricism is forbidden in mathematics. There are ontologies for mathematics that compete with formalism, such as Platonism, structuralism, and constructivism, but none of these ontologies would ever accept or introduce empiricism in mathematics. That is simply forbidden in all its ontologies.

According to the formalist ontology, mathematics is systematic string manipulation that is not "about" anything at all, and certainly not about the physical universe.

[Veritas Aequitas]

The best theory we have so far is the scientific Big Bang where the origin is "physical" as defined below.

The Big Bang model describes some of the early history of the universe but not its cause. Where is the reproducible experimental test report for the cause of the Big Bang? When was this experimental test report produced and when reproduced?

Science is not metaphysics, i.e. chasing after an ontological cause.
Science at best study as far as the evidences support its conclusion.
So far, the empirical evidence support the theory of the Big Bang where all the physical [as defined] reality begins.
There is nothing non-physical to be concluded.

The scientific model [& mathematic] is the most reliable at present.

For most questions about the physical universe, the answer cannot be justified with a reproducible experimental test report.

As I had stated there are different methods in arriving at scientific conclusion with different degree of credibility and reliability and they are the best we have at present.
Where there is no reproducible experiment can be done, e.g. the Big Bang, the conclusion is still reliable subject to that limitation.
While the Big Bang is not reproducible, various empirical evidences of the universe aligns with the theory of the Big Bang. There is no better theory to explain the current events in the Universe.
Do you have a better theory than the Big Bang [with its limitations]?

If you don't agree what non-scientific model are you relying upon.

You do not have an experimental test report to justify anything you say about the subject, simply, because such experimental test report does not even exist.

Note my question,
If you don't agree what non-scientific model are you relying upon.

I had pointed out what is mathematics [regardless of whatever the formalism] it is ultimately reducible to the empirical and the physical on a priori basis.

This view is wrong. Mathematics is ultimately reducible to systematic string manipulation, and nothing else. Furthermore, empiricism is forbidden in mathematics, just like it is forbidden in all branches of Pure Reason.

Note Kant's argument on how Science and Mathematics is possible on a synthetic a priori basis, while metaphysics [ontological] is not possible.

Incorrect about science.

Science is absolutely not divorced from sensory input and is therefore not Pure Reason at all. Hence, in Kantian lingo, science is not synthetic a priori.

Regardless of whatever the mathematics, it is reducible to the empirical a priori.

You are in denial of David Hilbert's formalist conclusions on the matter. You are in denial of one of the core ontological principles in mathematics: empiricism is forbidden in mathematics. There are ontologies for mathematics that compete with formalism, such as Platonism, structuralism, and constructivism, but none of these ontologies would ever accept or introduce empiricism in mathematics. That is simply forbidden in all its ontologies.

According to the formalist ontology, mathematics is systematic string manipulation that is not "about" anything at all, and certainly not about the physical universe.

Regardless of what you stated, the ultimate origin of mathematics is leveraged on the empirical as I had pointed out in a link earlier.
If it is not abstracted via reason from the empirical, then where does it comes from?

[godelian]

Science is not metaphysics, i.e. chasing after an ontological cause.
Science at best study as far as the evidences support its conclusion.
So far, the empirical evidence support the theory of the Big Bang where all the physical [as defined] reality begins.
There is nothing non-physical to be concluded.

Indeed. Therefore, science is not the right tool to answer questions about the first cause.

Do you have a better theory than the Big Bang [with its limitations]?

No. I do not have a reproducible experimental test report that suggests an alternative. But then again, I am personally not even interested in any models for the early history of the universe. I do not seek to read up on research on the matter, and I do not seek to do original research on the matter either.

Note my question,
If you don't agree what non-scientific model are you relying upon.

We already concluded that experimental testing is not the right tool for this problem.
My views on the first cause are necessarily foundationalist/axiomatic.

Regardless of what you stated, the ultimate origin of mathematics is leveraged on the empirical as I had pointed out in a link earlier.
If it is not abstracted via reason from the empirical, then where does it comes from?

Some subdivisions of mathematics, such as classical geometry and very basic arithmetic were originally developed by the Babylonians and Egyptians for the purpose of warehouse management, inventory control, and land taxation. Proper mathematics as an axiomatic discipline as we know it today, only started later on, with the publication of Euclid's elements. The old approach of informal mathematics was thereby abandoned.

Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics.

Informal mathematics means any informal mathematical practices, as used in everyday life, or by aboriginal or ancient peoples, without historical or geographical limitation. Modern mathematics, exceptionally from that point of view, emphasizes formal and strict proofs of all statements from given axioms.

Several ancient societies built impressive mathematical systems and carried out complex calculations based on proofless heuristics and practical approaches. Mathematical facts were accepted on a pragmatic basis. Empirical methods, as in science, provided the justification for a given technique.

So, indeed, aboriginals use empirical notions of mathematics.

The impetus for the axiomatic method and reformulating mathematics as a foundationalist discipline arises in Aristotle's publication "Posterior Analytics":

Aristotle's discussions on the best format for a deductive science in the Posterior Analytics reflect the practice of contemporary mathematics as taught and practiced in Plato's Academy, discussions there about the nature of mathematical sciences, and Aristotle's own discoveries in logic. Aristotle has two separate concerns. One evolves from his argument that there must be first, unprovable principles for any science, in order to avoid both circularity and infinite regresses. Aristotle distinguishes (Posterior Analytics i.2) Two sorts of starting points for demonstration, axioms and posits.

It is the understanding of Aristotelian foundationalism that allowed Euclid to rephrase and restructure existing aboriginal geometry -- which was formally not truly mathematics -- into a formal axiomatic system:

Aristotle and First Principles in Greek Mathematics. It has long been a tradition to read Aristotle's treatment of first principles as reflected in the first principles of Euclid's Elements I. There are similarities and differences. Euclid divides his principles into Definitions (horoi), Postulates (aitêmata), and Common Notions (koinai ennoiai).

I reject pre-Aristotelian aboriginal empiricism as merely some bags full of useful and handy tricks that are mostly unrelated to modern mathematics.

Pure Reason is blind, because it rejects sensory input. Pure Reason is always foundationalist.

The construction rules for a formal system are never justifiable by the system itself. Hence, you can consider such construction axioms to be "faith", if you want. Hence, Pure Reason must indeed derive all its conclusions from the "blind faith" in the construction rules of the system. Without "blind faith", there is no Pure Reason. Without "blind faith", there is no modern Aristotelian mathematics. The essence of foundationalism is "blind faith" in First Principles, i.e. its basic beliefs:

Identifying the alternatives as either circular reasoning or infinite regress, and thus exhibiting the regress problem, Aristotle made foundationalism his own clear choice, positing basic beliefs underpinning others.

Your way of thinking about mathematics, i.e. empiricism, is a throwback to pre-Aristotelian times. Your views are rejected in modern mathematics. They belong to the ethno-cultural studies of aboriginal tribes.

[Veritas Aequitas]

Science is not metaphysics, i.e. chasing after an ontological cause.
Science at best study as far as the evidences support its conclusion.
So far, the empirical evidence support the theory of the Big Bang where all the physical [as defined] reality begins.
There is nothing non-physical to be concluded.

Indeed. Therefore, science is not the right tool to answer questions about the first cause.

Do you have a better theory than the Big Bang [with its limitations]?

No. I do not have a reproducible experimental test report that suggests an alternative. But then again, I am personally not even interested in any models for the early history of the universe. I do not seek to read up on research on the matter, and I do not seek to do original research on the matter either.

Note my question,
If you don't agree what non-scientific model are you relying upon.

We already concluded that experimental testing is not the right tool for this problem.
My views on the first cause are necessarily foundationalist/axiomatic.

Note, you still have to justify your argument which you have not.

• Differing foundations may reflect differing epistemological emphases—empiricists emphasizing experience, rationalists emphasizing reason—but may blend both.
  https://en.wikipedia.org/wiki/Foundationalism

As Kant had demonstrated, foundationalism based on pure reason leads to illusions.

Regardless of what you stated, the ultimate origin of mathematics is leveraged on the empirical as I had pointed out in a link earlier.
If it is not abstracted via reason from the empirical, then where does it comes from?

Some subdivisions of mathematics, such as classical geometry and very basic arithmetic were originally developed by the Babylonians and Egyptians for the purpose of warehouse management, inventory control, and land taxation. Proper mathematics as an axiomatic discipline as we know it today, only started later on, with the publication of Euclid's elements. The old approach of informal mathematics was thereby abandoned.

Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics.

Informal mathematics means any informal mathematical practices, as used in everyday life, or by aboriginal or ancient peoples, without historical or geographical limitation. Modern mathematics, exceptionally from that point of view, emphasizes formal and strict proofs of all statements from given axioms.

Several ancient societies built impressive mathematical systems and carried out complex calculations based on proofless heuristics and practical approaches. Mathematical facts were accepted on a pragmatic basis. Empirical methods, as in science, provided the justification for a given technique.

So, indeed, aboriginals use empirical notions of mathematics.

The impetus for the axiomatic method and reformulating mathematics as a foundationalist discipline arises in Aristotle's publication "Posterior Analytics":

Aristotle's discussions on the best format for a deductive science in the Posterior Analytics reflect the practice of contemporary mathematics as taught and practiced in Plato's Academy, discussions there about the nature of mathematical sciences, and Aristotle's own discoveries in logic. Aristotle has two separate concerns. One evolves from his argument that there must be first, unprovable principles for any science, in order to avoid both circularity and infinite regresses. Aristotle distinguishes (Posterior Analytics i.2) Two sorts of starting points for demonstration, axioms and posits.

It is the understanding of Aristotelian foundationalism that allowed Euclid to rephrase and restructure existing aboriginal geometry -- which was formally not truly mathematics -- into a formal axiomatic system:

Aristotle and First Principles in Greek Mathematics. It has long been a tradition to read Aristotle's treatment of first principles as reflected in the first principles of Euclid's Elements I. There are similarities and differences. Euclid divides his principles into Definitions (horoi), Postulates (aitêmata), and Common Notions (koinai ennoiai).

I reject pre-Aristotelian aboriginal empiricism as merely some bags full of useful and handy tricks that are mostly unrelated to modern mathematics.

Pure Reason is blind, because it rejects sensory input. Pure Reason is always foundationalist.

The construction rules for a formal system are never justifiable by the system itself. Hence, you can consider such construction axioms to be "faith", if you want. Hence, Pure Reason must indeed derive all its conclusions from the "blind faith" in the construction rules of the system. Without "blind faith", there is no Pure Reason. Without "blind faith", there is no modern Aristotelian mathematics. The essence of foundationalism is "blind faith" in First Principles, i.e. its basic beliefs:

Identifying the alternatives as either circular reasoning or infinite regress, and thus exhibiting the regress problem, Aristotle made foundationalism his own clear choice, positing basic beliefs underpinning others.

Your way of thinking about mathematics, i.e. empiricism, is a throwback to pre-Aristotelian times. Your views are rejected in modern mathematics. They belong to the ethno-cultural studies of aboriginal tribes.

What??
Basic arithmetic with plus +, minus -, times x, divide / and "=" are still embedded in the most advanced modern mathematics.
As I had shown, the principles of basic arithmetic are abstracted from the empirical.

Note the axioms of Geometry are abstracted from observable empirical shapes [circles, triangles, square, polygons] that exist everywhere.
It is the same with basic arithmetic.

[godelian]

As Kant had demonstrated, foundationalism based on pure reason leads to illusions.

Kant uses the term "illusion" for the modern concept of "abstraction". Therefore, we can indeed say that foundationalism based on pure reason leads to "abstractions".

Basic arithmetic with plus +, minus -, times x, divide / and "=" are still embedded in the most advanced modern mathematics.

The definitions for addition (and multiplication) in Peano Arithmetic Theory have nothing to do with aboriginal empiricism.

Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as:

a + 0 = a [1]
a + S ( b ) = S ( a + b ) [2]

What does this definition have to do with the finger counting practices of aboriginal tribes?

As I had shown, the principles of basic arithmetic are abstracted from the empirical.

No, not at all. The recursive definition of addition cannot be obtained from aboriginal empiricism.

Note the axioms of Geometry are abstracted from observable empirical shapes [circles, triangles, square, polygons] that exist everywhere.
It is the same with basic arithmetic.

Classical Euclidean geometry, i.e. Euclid's publication called "Elements", was indeed the rephrasing of Egyptian land accounting tricks and rules of thumb onto a more solid foundation of axiomatic postulates. But then again, classical Euclidean geometry has long been abandoned in modern mathematics and replaced by algebraic geometry.

The classical Euclidean operations of straightedge-and-compass can only produce constructible numbers:

In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots.

Modern mathematics is no longer just about constructible numbers. We use far more complex numbers than that. The old Euclidean axiomatization in terms compass-and-straightedge operations is nowadays merely a historical curiosum. Modern algebraic geometry is pure symbol manipulation. It has nothing to do with empiricism.

[Veritas Aequitas]

As Kant had demonstrated, foundationalism based on pure reason leads to illusions.

Kant uses the term "illusion" for the modern concept of "abstraction". Therefore, we can indeed say that foundationalism based on pure reason leads to "abstractions".

From where did you arrive at the above?
I am a reasonably expert on Kantian philosophy and you are obviously wrong on the above.
The 'illusion' referred here, as I had mentioned is in particular to 'god' which is an illusion but theists reified such an illusion to be real.

Basic arithmetic with plus +, minus -, times x, divide / and "=" are still embedded in the most advanced modern mathematics.

The definitions for addition (and multiplication) in Peano Arithmetic Theory have nothing to do with aboriginal empiricism.

Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as:

a + 0 = a [1]
a + S ( b ) = S ( a + b ) [2]

What does this definition have to do with the finger counting practices of aboriginal tribes?

As I had shown, the principles of basic arithmetic are abstracted from the empirical.

No, not at all. The recursive definition of addition cannot be obtained from aboriginal empiricism.

Note the axioms of Geometry are abstracted from observable empirical shapes [circles, triangles, square, polygons] that exist everywhere.
It is the same with basic arithmetic.

Classical Euclidean geometry, i.e. Euclid's publication called "Elements", was indeed the rephrasing of Egyptian land accounting tricks and rules of thumb onto a more solid foundation of axiomatic postulates. But then again, classical Euclidean geometry has long been abandoned in modern mathematics and replaced by algebraic geometry.

The classical Euclidean operations of straightedge-and-compass can only produce constructible numbers:

In terms of algebra, a length is constructible if and only if it represents a constructible number, and an angle is constructible if and only if its cosine is a constructible number. A number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots.

Modern mathematics is no longer just about constructible numbers. We use far more complex numbers than that. The old Euclidean axiomatization in terms compass-and-straightedge operations is nowadays merely a historical curiosum. Modern algebraic geometry is pure symbol manipulation. It has nothing to do with empiricism.

As I had shown, the principles of basic arithmetic are abstracted from the empirical and these are the fundamental roots of all modern mathematics.

For example you may have say a number like 0.00000000000000000000000000001 in modern mathematics which appear to be complex but note the "1" at the end of it.
This is reducible to the empirical concept of 1 'finger' or 'toes' in relation to a single unit of empirical entity.

[godelian]

I am a reasonably expert on Kantian philosophy and you are obviously wrong on the above.
The 'illusion' referred here, as I had mentioned is in particular to 'god' which is an illusion but theists reified such an illusion to be real.

You make Kant sound like an atheist, while he clearly wasn't.

Near the beginning of his Universal Natural History and Theory of the Heavens of 1755, Kant observes that the harmonious order of the universe points to its divinely governing first Cause; near the end of it, he writes that even now the universe is permeated by the divine energy of an omnipotent Deity (Cosmogony, pp. 14 and 153). In his New Exposition of the First Principles of Metaphysical Knowledge (of the same year), he points to God’s existence as the necessary condition of all possibility (Exposition, pp. 224-225).

As I had shown, the principles of basic arithmetic are abstracted from the empirical and these are the fundamental roots of all modern mathematics.
For example you may have say a number like 0.00000000000000000000000000001 in modern mathematics which appear to be complex but note the "1" at the end of it.
This is reducible to the empirical concept of 1 'finger' or 'toes' in relation to a single unit of empirical entity.

As I mentioned before, counting fingers and toes is the basis for aboriginal mathematics, which is indeed empirical.

Aristotle did not want that.
Euclid did not want that.

It is the classical Greek rejection of and disdain for empiricism in knowledge that led to purely axiomatic mathematics, i.e. Pure Reason.

Modern mathematics is a complete replacement and repudiation of the aboriginal empiricism that was in use before modern mathematics.

Modern mathematics is Pure Reason that has blind faith in First Principles.

It is actually still possible to use the informal pre-Aristotelian mathematics of aboriginal tribes. Their methods are not completely incorrect. The tricks often work. However, modern science does not make use of aboriginal mathematics but undeniably prefers modern mathematics.

[Veritas Aequitas]

I am a reasonably expert on Kantian philosophy and you are obviously wrong on the above.
The 'illusion' referred here, as I had mentioned is in particular to 'god' which is an illusion but theists reified such an illusion to be real.

You make Kant sound like an atheist, while he clearly wasn't.

Kant is a Deist and abhor the idea of a personal god of the theists.

Even then he is likely a closet non-theist given that he was reprimanded by the King when he critiqued religion and theism severely and his tenure as a professor was at stake.

• In October 1794, Kant received a royal rescript from the court of Frederick William II reprimanding him for his heterodox writings on Christianity and prohibiting him from further publication which “distort and disparage” Christianity (cf. AK 7:6 [1798]).
  https://plato.stanford.edu/entries/kant-religion/

Near the beginning of his Universal Natural History and Theory of the Heavens of 1755, Kant observes that the harmonious order of the universe points to its divinely governing first Cause; near the end of it, he writes that even now the universe is permeated by the divine energy of an omnipotent Deity (Cosmogony, pp. 14 and 153). In his New Exposition of the First Principles of Metaphysical Knowledge (of the same year), he points to God’s existence as the necessary condition of all possibility (Exposition, pp. 224-225).

The early-Kant was pro-God but not the later-Kant who turned Deist due to the predominant theistic pressure during his time.

• Kant has long been seen as hostile to religion. Many of his contemporaries, ranging from his students to the Prussian authorities, saw his Critical project as inimical to traditional Christianity.
  The impression of Kant as a fundamentally secular philosopher became even more deeply entrenched through the twentieth century, though this is belied by a closer inspection of his writings both before and after the publication of his Critique of Pure Reason (1781), i.e., what are commonly referred to as his “pre-Critical” and “Critical” periods.
  https://plato.stanford.edu/entries/kant-religion/

As I had shown, the principles of basic arithmetic are abstracted from the empirical and these are the fundamental roots of all modern mathematics.
For example you may have say a number like 0.00000000000000000000000000001 in modern mathematics which appear to be complex but note the "1" at the end of it.
This is reducible to the empirical concept of 1 'finger' or 'toes' in relation to a single unit of empirical entity.

As I mentioned before, counting fingers and toes is the basis for aboriginal mathematics, which is indeed empirical.

Aristotle did not want that.
Euclid did not want that.

It is the classical Greek rejection of and disdain for empiricism in knowledge that led to purely axiomatic mathematics, i.e. Pure Reason.

Modern mathematics is a complete replacement and repudiation of the aboriginal empiricism that was in use before modern mathematics.

Modern mathematics is Pure Reason that has blind faith in First Principles.

It is actually still possible to use the informal pre-Aristotelian mathematics of aboriginal tribes. Their methods are not completely incorrect. The tricks often work. However, modern science does not make use of aboriginal mathematics but undeniably prefers modern mathematics.

You don't seem to grasp the point.
I am not insisting modern mathematicians resorts to finger and toes in doing their maths.
This would be very stupid.
What I meant their mathematics are reducible [traceable] ultimately to their roots in the empirical.

Even now we are still relying on fingers to teach children as parents and in nursery schools. This is will be the foundation that would be embedded deep into their consciousness while they proceeded to more sophisticated arithmetic and mathematics as the grow older.

I believe no human would be able to proceed to do the most complex modern mathematics without the embedded empirical roots of mathematics.

[Skepdick]

Modern mathematics is no longer just about constructible numbers. We use far more complex numbers than that. The old Euclidean axiomatization in terms compass-and-straightedge operations is nowadays merely a historical curiosum. Modern algebraic geometry is pure symbol manipulation. It has nothing to do with empiricism.

The idea of what it means to "construct" a mathematical object has evolved with the times. We don't use a straight edge and a compass now - a computer will do. From the lens of constructivism it's not enough to say that a Mathematical object exist - it's no longer enough to assume non-existence and derrive a contradiction. One must also produce an example of such object as evidence for its existence. This is the BHK interpretation (realisability interpretation) or thereabouts. In so far as evidence/witnessing of existence is necessary constructivism is empirical.

Including Algebraic Geometry.

https://www.amazon.com/Algebraic-Geomet ... 0367398966

Just about every branch of Mathematics viewed from the constructivist lens is constructive (and therefore empirical) and the more the field is mechanized the less it resembles axiomatics; and the more it resembles reverse mathematics: choose a theorem and search for sufficient justifications (axioms).

To cut a long story short - Mathematics is about notation, not denotation. Syntax, not semantics. The semantic view of Mathematics is a religion.

[godelian]

Even now we are still relying on fingers to teach children as parents and in nursery schools. This is will be the foundation that would be embedded deep into their consciousness while they proceeded to more sophisticated arithmetic and mathematics as the grow older.

I believe no human would be able to proceed to do the most complex modern mathematics without the embedded empirical roots of mathematics.

One reason why so many people are lousy at modern mathematics is because they cannot let go of the empirical aboriginal basic mathematics that they learned during childhood. They think that it is the same thing, or should be the same thing, while it isn't.

In fact, these people never learn to deal with pure abstraction or pure reason, because they always keep looking for some comparable analog in the physical universe while there isn't one.

How can you explain to someone Abel-Ruffini theorem, i.e. that the quintic is not generally solvable in radicals, if that person keeps looking for physical objects that would resemble a quintic? There simply isn't one.

[Skepdick]

One reason why so many people are lousy at modern mathematics is because they cannot let go of the empirical aboriginal basic mathematics that they learned during childhood. They think that it is the same thing, or should be the same thing, while it isn't.

In fact, these people never learn to deal with pure abstraction or pure reason, because they always keep looking for some comparable analog in the physical universe while there isn't one.

How can you explain to someone Abel-Ruffini theorem, i.e. that the quintic is not generally solvable in radicals, if that person keeps looking for physical objects that would resemble a quintic? There simply isn't one.

Ohhh, you sound like a set theorist. The notion of "purity" is entirely bogus.

If Mathematics is "just" symbol manipulation then the analog in the physical universe is programming. Symbolic logic. A proof is a terminating decision procedure.

Can you define a "quintic" in a theorem prover?
Can I interact with it?
Examine how it behaves? Examine its source code? Examine how it interacts with other objects in the universe of discourse?
If I can learn what a quintic is - It's empirical. How else could I learn it otherwise?

When you have implemented a quintic in a theorem prover the physical analog of a quintic would be the physical configuration of the computer on which it's implemented.

Of course, the construction may be very complex and have minutae of important and relevant details, but ultimately it's teachable to a computer; and learnable by a human - so it's empirical.

[godelian]

In so far as evidence/witnessing of existence is necessary constructivism is empirical.

A witness is just a symbolic example that satisfies a particular set membership function. There is absolutely nothing empirical about producing such symbol string.

Example: there are numbers divisible by 3 and by 5.

Answer: Yes, 45 is a witness.

The symbol "45" is a legitimate answer to this constructivist question but has absolutely nothing empirical about it. What would there be empirical about any natural number or any mathematical object for that matter?

[Skepdick]

A witness is just a symbolic example that satisfies a particular set membership function. There is absolutely nothing empirical about producing such symbol string.

The decision procedure (classification?) of whether any particular string belongs to any particular language is language recognition. Which is what you are doing right now.

Is this sentence a member of the English language?
Is producing English symbols an empirical endeavor?

It sure seems like it - I am generating them and you are parsing/lexing them.

Example: there are numbers divisible by 3 and by 5.

Answer: Yes, 45 is a witness.

The symbol "45" is a legitimate answer to this constructivist question but has absolutely nothing empirical about it. What would there be empirical about any natural number or any mathematical object for that matter?

The fact that you can explain the verification procedure makes it empirical and 100% mechnical.
The fact that you can produce an algorithm which will mechanically produce all the numbers which satisfy the condition makes it 100% empirical.
Obviously the configuration of the electrical circuitry is messy and complicated, but it's empirical.
It's a physical machine which produces physical output.

And since we are talking about satisfiability we are necessarily in the domain of SAT solvers because when it comes to powersets and large cardinals all you are doing is Infinitary combinatorics. It's all just brute force and searching.

Completely unrelated aside. You think 45 and "45" are the same type of thing. Lol! That's a type error. Of course, you probably don't know what those are - you don't have effect systems or side effects in "pure mathematics". So you literally can't distinguish between a function which computes 1+1 but keeps the result to itself...

Code: Select all

def function():
   1+1

and a function which computes 1+1 and communicates the result to any observer.

Code: Select all

def function():
   return 1+1

The former is a "pure function" - a black hole.
The latter is empirical.

[Veritas Aequitas]

Even now we are still relying on fingers to teach children as parents and in nursery schools. This is will be the foundation that would be embedded deep into their consciousness while they proceeded to more sophisticated arithmetic and mathematics as the grow older.

I believe no human would be able to proceed to do the most complex modern mathematics without the embedded empirical roots of mathematics.

One reason why so many people are lousy at modern mathematics is because they cannot let go of the empirical aboriginal basic mathematics that they learned during childhood. They think that it is the same thing, or should be the same thing, while it isn't.

In fact, these people never learn to deal with pure abstraction or pure reason, because they always keep looking for some comparable analog in the physical universe while there isn't one.

How can you explain to someone Abel-Ruffini theorem, i.e. that the quintic is not generally solvable in radicals, if that person keeps looking for physical objects that would resemble a quintic? There simply isn't one.

You missed my point.

I have never implied the following;

In fact, these people never learn to deal with pure abstraction or pure reason, because they always keep looking for some comparable analog in the physical universe while there isn't one.

What I am referring to is the Philosophy of Mathematics, i.e. upon reflection all modern mathematics are reducible to its roots in the empirical, e.g. numbers which are abstracted from the human empirical sense.

Note how your supposedly complex quintic is eventually translated to the empirical as below. Where numbers are used, their origin was to the empirical fingers or some single entity.

Application to celestial mechanics
Solving for the locations of the Lagrangian points of an astronomical orbit in which the masses of both objects are non-negligible involves solving a quintic.

More precisely, the locations of L2 and L1 are the solutions to the following equations, where the gravitational forces of two masses on a third (for example, Sun and Earth on satellites such as Gaia and the James Webb Space Telescope at L2 and SOHO at L1) provide the satellite's centripetal force necessary to be in a synchronous orbit with Earth around the Sun:

The ± sign corresponds to L2 and L1, respectively; G is the gravitational constant, ω the angular velocity, r the distance of the satellite to Earth, R the distance Sun to Earth (that is, the semi-major axis of Earth's orbit), and m, ME, and MS are the respective masses of satellite, Earth, and Sun.
https://en.wikipedia.org/wiki/Quintic_f ... _mechanics

Note distances were originally empirical as well.
E.g. the foot was with reference to the 'length' of someone's foot.
one [one finger] foot [someone's foot].

Then the meter is a standard measurement of length based on something empirical,
https://en.wikipedia.org/wiki/Metre

The metre is currently defined as the length of the path travelled by light in a vacuum in
1 / 299 792 458 of a second.

The metre was originally defined
in 1793 as one ten-millionth of the distance from the equator to the North Pole along a great circle, so the Earth's circumference is approximately 40000 km.
In 1799, the metre was redefined in terms of a prototype metre bar (the actual bar used was changed in 1889).
In 1960, the metre was redefined in terms of a certain number of wavelengths of a certain emission line of krypton-86.
The current definition was adopted in 1983 and modified slightly in 2002 to clarify that the metre is a measure of proper length.

Thus my point again, the fundamentals of mathematics [old or new] is reducible to the empirical.

Even if you can come up with illusory mathematics via pure reason or whatever, the fundamental is still empirical.

For example theists invent an illusory God but basically such an invented entity has anthropomorphic [empirical] qualities.
It is impossible for an absolutely non-empirical thing like a square-circle to exists.

[Skepdick]

What I am referring to is the Philosophy of Mathematics, i.e. upon reflection all modern mathematics are reducible to its roots in the empirical

Mathematics is reducible to reduction.

https://en.wikipedia.org/wiki/Reduction_(complexity)