3 I explicitly added that applying mathematics to the real world is adding a new rule, and improved the analogy in conversational language.
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I agree with jobermark's implication that physics and other sciences are more than the pure mathematics they employ. The object of science is usually not to describe imagined systems but to describe real, physical systems and objects. As such it is common for a scientist to say, not only is this a valid mathematical statement but it also is a useful statement that describes the real world! This would be a new rule which introduces falsifiability, since presumably the scientist did not add a complete and accurate set of rules to precisely describe the real world.

I would compare this misapplication of mathematics to using a false analogy in conversation with any other language. The language itself is perfectly valid, and one could indeed imagine a universe where a sentence such as "I am seventynine feet tall" is true. ButIt may get the point across that I am a tall person, but it makes no sensewould be misleading (an illusion) to usepresent that sort of language to describe realitysentence as true when I am not actually nine feet tall.

I agree with jobermark's implication that physics and other sciences are more than the pure mathematics they employ. The object of science is usually not to describe imagined systems but to describe real, physical systems and objects. As such it is common for a scientist to say, not only is this a valid mathematical statement but it also is a useful statement that describes the real world!

I would compare this misapplication of mathematics to using a false analogy in conversation with any other language. The language itself is perfectly valid, and one could indeed imagine a universe where a sentence such as "I am seventy feet tall" is true. But it makes no sense to use that sort of language to describe reality.

I agree with jobermark's implication that physics and other sciences are more than the pure mathematics they employ. The object of science is usually not to describe imagined systems but to describe real, physical systems and objects. As such it is common for a scientist to say, not only is this a valid mathematical statement but it also is a useful statement that describes the real world! This would be a new rule which introduces falsifiability, since presumably the scientist did not add a complete and accurate set of rules to precisely describe the real world.

I would compare this misapplication of mathematics to using a false analogy in conversation with any other language. The language itself is perfectly valid, and one could indeed imagine a universe where a sentence such as "I am nine feet tall" is true. It may get the point across that I am a tall person, but it would be misleading (an illusion) to present that sentence as true when I am not actually nine feet tall.

2 I added responses to jobermark's comment.
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Responses to comments

@jobermark has expressed concern that my definition of mathematics would apply to physics and other basic sciences, whereas he asserts these fields are not mathematics but only make use of mathematics.

I do believe the formulas of physics and other sciences, if expressed in mathematics, are in fact a form of mathematics. I stand by my opinion that no valid mathematical statement - even in the context of physics or another science - can be an illusion in its own right.

I agree with jobermark's implication that physics and other sciences are more than the pure mathematics they employ. The object of science is usually not to describe imagined systems but to describe real, physical systems and objects. As such it is common for a scientist to say, not only is this a valid mathematical statement but it also is a useful statement that describes the real world!

Sometimes the mathematics used, while valid on its own, just doesn't apply to the situation as advertised. A physicist of old may have said, velocity is the product of speed and time. It is implied that this relation applies to reality, not just some imagined universe. A physicist today may retort, this equation does not hold at speeds approaching c. We have conducted experiments and gathered evidence in the real world which contradicts that formula's applicability to reality.

I would compare this misapplication of mathematics to using a false analogy in conversation with any other language. The language itself is perfectly valid, and one could indeed imagine a universe where a sentence such as "I am seventy feet tall" is true. But it makes no sense to use that sort of language to describe reality.

@jobermark also expresses concern that my definition of illusion excludes any successful illusions, for example alchemical homeopathy.

For similar reasons to the above, I stand by my definition. Aelius Galenus may have proposed the relation, an excess of blood is positively correlated with optimism. Indeed such an equation could be valid in mathematical terms (optimism = blood + 6 liters, for example), but were Galen to assert that this relation applies to real people's blood and attitude (as he did), then that claim becomes falsifiable. The mathematics itself is not the illusion, only the application.

If jobermark meant to say that a valid mathematical statement might turn out to be an illusion because the rules change, I would say this is precluded by my definition of mathematics as following consistent rules. I don't think this was the intended argument, though.


Responses to comments

@jobermark has expressed concern that my definition of mathematics would apply to physics and other basic sciences, whereas he asserts these fields are not mathematics but only make use of mathematics.

I do believe the formulas of physics and other sciences, if expressed in mathematics, are in fact a form of mathematics. I stand by my opinion that no valid mathematical statement - even in the context of physics or another science - can be an illusion in its own right.

I agree with jobermark's implication that physics and other sciences are more than the pure mathematics they employ. The object of science is usually not to describe imagined systems but to describe real, physical systems and objects. As such it is common for a scientist to say, not only is this a valid mathematical statement but it also is a useful statement that describes the real world!

Sometimes the mathematics used, while valid on its own, just doesn't apply to the situation as advertised. A physicist of old may have said, velocity is the product of speed and time. It is implied that this relation applies to reality, not just some imagined universe. A physicist today may retort, this equation does not hold at speeds approaching c. We have conducted experiments and gathered evidence in the real world which contradicts that formula's applicability to reality.

I would compare this misapplication of mathematics to using a false analogy in conversation with any other language. The language itself is perfectly valid, and one could indeed imagine a universe where a sentence such as "I am seventy feet tall" is true. But it makes no sense to use that sort of language to describe reality.

@jobermark also expresses concern that my definition of illusion excludes any successful illusions, for example alchemical homeopathy.

For similar reasons to the above, I stand by my definition. Aelius Galenus may have proposed the relation, an excess of blood is positively correlated with optimism. Indeed such an equation could be valid in mathematical terms (optimism = blood + 6 liters, for example), but were Galen to assert that this relation applies to real people's blood and attitude (as he did), then that claim becomes falsifiable. The mathematics itself is not the illusion, only the application.

If jobermark meant to say that a valid mathematical statement might turn out to be an illusion because the rules change, I would say this is precluded by my definition of mathematics as following consistent rules. I don't think this was the intended argument, though.

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There are three questions here. I will pass over the first question (whether any philosopher made the claim) out of ignorance.

The second question I crafted from the original post: How can we know that mathematics cannot be an illusion created by the mind?

My answer to this question depends on the definitions of the terms, in context. I can offer an answer of sorts, but some people may necessarily object to my definitions. Some may object to my paraphrasing of the question above. I am new to the site and welcome your criticism.

I personally consider "mathematics", in the context of this question, to be any one in a family of languages with strict syntax and consistent semantic rules. The language has strict enough syntax that any grammatical statement is unambiguous. The rules are consistent in the sense that rules don't contradict each other and no valid statement will ever contradict a rule; in elementary mathematics, "one times seven equals one" is always illegal because it contradicts an established rule (the multiplicative property).

I would consider an "illusion", in this context, to be a belief proven false by contradictions of evidence. For example the entasis and curved lines in some classical Greek architecture provide the illusion of straight lines: visually the columns and walls appear straight, but a more detailed measurement will show the curvature.

Now if mathematics is consistent, as it is by my definition, there will be no contradictions between rules and valid statements. Neither will there be contradictions between any two rules. Therefore no evidence of a contradiction can be found between a valid statement and a rule, nor between any two rules. Therefore neither mathematics nor a valid mathematical statement can be an illusion, as I define it.


The third question is ambiguous. I can't identify the subject or direct object for the last clause, which reads "... or is [what?] negatively affected by [which?] illusion?" I'm guessing it was supposed to be "is the mind negatively affected by the illusion of mathematics?" Regardless, I have asserted that mathematics is not an illusion so this third question would not apply.