The position of constructive mathematics on the axiom of infinity is outright unsustainable

[godelian]

The infinite repetition of operations is a central principle in algebra and calculus.

In fact, the very notion of "real number" is constructed by means of infinitely repeating operations (by both Dedekind and Cauchy).

You must assume that such infinitely repeating process is possible and that it will terminate. That clearly amounts to assuming that you can traverse the set of the natural numbers from beginning till end, which is exactly what the axiom of infinity proclaims.

Constructive mathematics can criticize the axiom of infinity all they want, but since they do not have an alternative to algebra and calculus, I do not think that anybody will want to listen to them.

Seriously, the axiom of infinity is here to stay.

[Impenitent]

I had a thought...

even if the universe is expanding, isn't there a particular number of particles (molecules, atoms, subatomic particles- whatever) within it?

-Imp

[godelian]

I had a thought...

even if the universe is expanding, isn't there a particular number of particles (molecules, atoms, subatomic particles- whatever) within it?

-Imp

Not sure, actually.

In its essence, mathematics is not about the physical universe. It is about abstract Platonic objects in the universe of seemingly consistent abstraction.

In the Platonic world, there are countably and uncountably infinite sets and even collections that you are not even allowed to themselves put inside a collection.

The axiom of infinity is about the smallest possible infinite set, i.e. the one that contains the natural numbers. It is a standard belief in classical mathematics.

Both the constructivists and the finitists object to this axiom. I sometimes try to investigate their objections, but I usually end up rejecting them. The cost of adopting their point of view is very high, while I really don't see any tangible benefits.

[socrattus]

The axiom of infinity in mathematics is true because “infinity” appears in many physical processes.
Infinities.
Scientific motto: "If you get infinity (∞) in an equation it's usually a sign that you've made a mistake".
To avoid "a mistake" a mathematical method of renormalization must be used,
which according to Feynman is a method “to sweep the dust under the carpet".
#
'' So we really do not know exactly what it is that we are assuming that gives us
the difficulty producing infinities. A nice problem! However, it turns out that it is possible
to sweep the infinities under the rug, by a certain crude skill,
and temporarily we are able to keep on calculating. '' /Richard Feynman/

[godelian]

The axiom of infinity in mathematics is true because “infinity” appears in many physical processes.
Infinities.
Scientific motto: "If you get infinity (∞) in an equation it's usually a sign that you've made a mistake".
To avoid "a mistake" a mathematical method of renormalization must be used,
which according to Feynman is a method “to sweep the dust under the carpet".
#
'' So we really do not know exactly what it is that we are assuming that gives us
the difficulty producing infinities. A nice problem! However, it turns out that it is possible
to sweep the infinities under the rug, by a certain crude skill,
and temporarily we are able to keep on calculating. '' /Richard Feynman/

Science is about the physical universe. Mathematics is absolutely not. Feynman was a scientist. Mathematics is about the abstract Platonic universe of mathematical objects where infinite cardinalities are not errors at all, but perfectly fine first-class objects.

[socrattus]

The axiom of infinity in mathematics is true because “infinity” appears in many physical processes.
Infinities.
Scientific motto: "If you get infinity (∞) in an equation it's usually a sign that you've made a mistake".
To avoid "a mistake" a mathematical method of renormalization must be used,
which according to Feynman is a method “to sweep the dust under the carpet".
#
'' So we really do not know exactly what it is that we are assuming that gives us
the difficulty producing infinities. A nice problem! However, it turns out that it is possible
to sweep the infinities under the rug, by a certain crude skill,
and temporarily we are able to keep on calculating. '' /Richard Feynman/

Science is about the physical universe. Mathematics is absolutely not. Feynman was a scientist. Mathematics is about the abstract Platonic universe of mathematical objects where infinite cardinalities are not errors at all, but perfectly fine first-class objects.

It is said: "Mathematics is the queen of science."
But it is a queen only when it marries physics (reality),
otherwise it is mad woman.

[godelian]

The axiom of infinity in mathematics is true because “infinity” appears in many physical processes.
Infinities.
Scientific motto: "If you get infinity (∞) in an equation it's usually a sign that you've made a mistake".
To avoid "a mistake" a mathematical method of renormalization must be used,
which according to Feynman is a method “to sweep the dust under the carpet".
#
'' So we really do not know exactly what it is that we are assuming that gives us
the difficulty producing infinities. A nice problem! However, it turns out that it is possible
to sweep the infinities under the rug, by a certain crude skill,
and temporarily we are able to keep on calculating. '' /Richard Feynman/

Science is about the physical universe. Mathematics is absolutely not. Feynman was a scientist. Mathematics is about the abstract Platonic universe of mathematical objects where infinite cardinalities are not errors at all, but perfectly fine first-class objects.

It is said: "Mathematics is the queen of science."
But it is a queen only when it marries physics (reality),
otherwise it is mad woman.

I like that mad woman as is. I am absolutely not interested in applied mathematics. I find applied mathematics to be philosophically uninteresting, even a bit boring.

[Impenitent]

mad woman? tired woman... she's trying to get to the end of the square root of three...

-Imp

[socrattus]

Science is about the physical universe. Mathematics is absolutely not. Feynman was a scientist. Mathematics is about the abstract Platonic universe of mathematical objects where infinite cardinalities are not errors at all, but perfectly fine first-class objects.

It is said: "Mathematics is the queen of science."
But it is a queen only when it marries physics (reality),
otherwise it is mad woman.

I like that mad woman as is. I am absolutely not interested in applied mathematics. I find applied mathematics to be philosophically uninteresting, even a bit boring.

Mathematics & dimensions.
We live in Descartes' 3D
Minkowski discovered 4D
Kaluza and Klein developed 5D
String physicists created 7D, 10D, 11D, MD ....
Physicist R. Lipin wrote about extra mathematical dimensions:
“Give me three parameters and I can fit the elephant.
With four I can make him wiggle his trunk...
with one more parameter the elephant will fly.”
Did you like this crazy abstract math?

[godelian]

Mathematics & dimensions.
We live in Descartes' 3D
Minkowski discovered 4D
Kaluza and Klein developed 5D
String physicists created 7D, 10D, 11D, MD ....
Physicist R. Lipin wrote about extra mathematical dimensions:
“Give me three parameters and I can fit the elephant.
With four I can make him wiggle his trunk...
with one more parameter the elephant will fly.”
Did you like this crazy abstract math?

Yeah, but elephants are still about the physical universe.

Hence, the idea is still "applied".

In a sense, "applied" math is actually not math.

Math is about provability while there is absolutely nothing provable about the physical universe, which is at best testable, if even.

There are no elephants in the abstract Platonic universe. An elephant is a physical object -- part of physical reality -- that somehow may make sense to a physicist but not to mathematician.

[socrattus]

Mathematics & dimensions.
We live in Descartes' 3D
Minkowski discovered 4D
Kaluza and Klein developed 5D
String physicists created 7D, 10D, 11D, MD ....
Physicist R. Lipin wrote about extra mathematical dimensions:
“Give me three parameters and I can fit the elephant.
With four I can make him wiggle his trunk...
with one more parameter the elephant will fly.”
Did you like this crazy abstract math?

Yeah, but elephants are still about the physical universe.

Hence, the idea is still "applied".

In a sense, "applied" math is actually not math.

Math is about provability while there is absolutely nothing provable about the physical universe, which is at best testable, if even.

There are no elephants in the abstract Platonic universe. An elephant is a physical object -- part of physical reality -- that somehow may make sense to a physicist but not to mathematician.

Conclusion:
elephants ... " somehow may make sense to a physicist but not to mathematician." ...
We have successful modern technology thanks to the math of quantum physics.
This means that the "elephants" must be real for the mathematician too.

[godelian]

elephants ... " somehow may make sense to a physicist but not to mathematician." ...
We have successful modern technology thanks to the math of quantum physics.
This means that the "elephants" must be real for the mathematician too.

No, because reality for a mathematician is not the physical universe. Reality is the abstract Platonic universe. If you can physically see, hear, or smell it, then it is not the real stuff. Physicalities are only Platonic shadows. They may be testable but there is absolutely nothing provable about them.

Pure reason is deaf and blind.

Quantum physics is physically observable. That guarantees that mathematics does not want to have anything to do with it.

[luberti]

I fully agree with godelian throughout this thread up to this point, if I correctly understood the points made.

Moreover, I am convinced (partly due to numerous interactions with such people) that people who oppose the axiom of infinity and advocate constructivism (as a philosophy of mathematics) simply did not understand some mathematical key concepts when they started doing it.

Often they are not willing or even capable of understand these concepts, either: The longer they defended their beliefs, the more they tend to fall into some fallacious "if I'm wrong, it hurts, therefore I'm right" behavioral patterns, which is very common for humans.
So it oftentimes isn't even about the pursuit of truth, but very much against it. Though, it might have initially been about seeking truth, but back then the person was just too foolish to grasp some basic concepts and too quick to publicly express premature beliefs.

Though, I should note that constructivism (a philosophy about mathematics) and constructive mathematics (an area of mathematics) are very different things. The latter can absolutely make sense, especially for verification or computer science, since it studies constructive systems in general. For example, constructive proofs (in contrast to non-constructive proofs) show how certain objects can be constructed, which helps with designing algorithms to compute such objects. Intuitionistic mathematics (the study of systems that were designed to follow philosophical nonsense) is just a small area of many constructive areas, as also suggested by the MSC2020.

[Eodnhoj7]

The infinite repetition of operations is a central principle in algebra and calculus.

In fact, the very notion of "real number" is constructed by means of infinitely repeating operations (by both Dedekind and Cauchy).

You must assume that such infinitely repeating process is possible and that it will terminate. That clearly amounts to assuming that you can traverse the set of the natural numbers from beginning till end, which is exactly what the axiom of infinity proclaims.

Constructive mathematics can criticize the axiom of infinity all they want, but since they do not have an alternative to algebra and calculus, I do not think that anybody will want to listen to them.

Seriously, the axiom of infinity is here to stay.

You do understand that a purely abstract entity exists only within the observer's head as their is no physical medium to share it through the senses?

[godelian]

You do understand that a purely abstract entity exists only within the observer's head

I disagree. I subscribe to Platonism:

https://plato.stanford.edu/entries/plat ... athematics

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

as their is no physical medium to share it through the senses?

The interest rate on the 10-year US treasury bill and the exchange rate of JPY/USD are pure abstractions. There is no physical medium to share these things through the senses. Still, they are very, very real. Even by the mainstream media, these things are considered much more relevant than most material objects. They do not just exist within the observer's head.

There is no physical medium to share a tariff through the senses. So, a tariff only exists in Donald Trump's head. Therefore, the Chinese should not waste their time negotiating about mere fictions ... but apparently, they do!