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You ask:

Does math admit(care about) the existence of an entity?

If you consider math a formal science, and the question of existence an ontological and therefore philosophical question, then its best to say that contemporary, working mathematicians tend not to care so much about what it means for abstract objects (SEP) to exist, and leave such questions to philosophers of mathematics, preferring instead to perform the existential quantification with symbols and leave the truth of the matter intuitionintuition (SEP). Historically, there was little distinction between philosophy and mathematics with the ancient Greeks like Pythagoras and Plato being both philosophers and mathematicians. But today, working mathematicians are interested in the science of math, whereas philosophers, particularly philosophers of mathematics, are more interested in the philosophical question.

Andrew Wiles is an example of a working mathematician, and Stewart Shapiro is an example of a philosopher of mathematics. Wiles is famous for his solution to Fermat's Last Theorem, and immersed himself for decades in the mathematical and logical sciences to find a proof related to the conjecture. On the other hand, Shapiro is famous for his philosophical theory which is about various questions and is called structuralism. Wiles dealt with language and formal systems that were distinctly mathematical and logical in character, whereas Shapiro's work deals with natural language that seeks to describe and explain and categorize the mathematical logic of working mathematicians. Where the two men might share knowledge is in the domain of discourse known as mathematical explanation (SEP).

A working mathematician looks for proof and provides mathematical explanation, whereas a philosopher of mathematics seeks to put the practice of mathematical explanation into a philosophical framework to explain how explanation works. The former might be understood as the language game that mathematicians play, and the latter as the language game philosophers play. Mathematicians do math, whereas philosophers are more interested in understanding in a greater context what exactly it is that mathematicians do and how it relates to other topics like science, language, and the mind.

Working mathematicians generally agree on axioms and syntax when exchanging ideas and focus on proof theory, where as philosophers are quite disputatious and are often looking to answer metaphysical, epistemological, or ontological questions that have little to do with mathematical proof. One of the fascinating debates in philosophy of logic and mathematics is whether or not mathematical structure is real or not. Another is whether it is objective and mind-independent. Frege was an antipsychologist and claimed that mathematics and logic were not dependent on the mind. Brouwer was an intuitionist (SEP) who disagreed and sent some working mathematicians down the path of mathematical constructivism as a technique in math. Plato argued there was an alternative realm. Shapiro argues that structures are real and objective. Lakoff argues math is nothing more than neural computation. A good, brief introduction to the philosophy of math is Philosophy of Mathematics (GB) by Linnebo.

So, does math care? Mathematics is a theory, and doesn't care, but the people who do it can roughly divided into two groups: those who don't, the working mathematicians who tend towards Platonism and realism based on their intuitions, and those who do care, mathematical philosophers who have a bewildering array of views about mathematical existence of entities. In this way, formal scientists are a lot like physical scientists. They often see such philosophical disputes as irrelevant and don't spend much time on the question. They simply declare in their papers and books things exist and move on.

You ask:

Does math admit(care about) the existence of an entity?

If you consider math a formal science, and the question of existence an ontological and therefore philosophical question, then its best to say that contemporary, working mathematicians tend not to care so much about what it means for abstract objects (SEP) to exist, and leave such questions to philosophers of mathematics, preferring instead to perform the existential quantification with symbols and leave the truth of the matter intuition. Historically, there was little distinction between philosophy and mathematics with the ancient Greeks like Pythagoras and Plato being both philosophers and mathematicians. But today, working mathematicians are interested in the science of math, whereas philosophers, particularly philosophers of mathematics, are more interested in the philosophical question.

Andrew Wiles is an example of a working mathematician, and Stewart Shapiro is an example of a philosopher of mathematics. Wiles is famous for his solution to Fermat's Last Theorem, and immersed himself for decades in the mathematical and logical sciences to find a proof related to the conjecture. On the other hand, Shapiro is famous for his philosophical theory which is about various questions and is called structuralism. Wiles dealt with language and formal systems that were distinctly mathematical and logical in character, whereas Shapiro's work deals with natural language that seeks to describe and explain and categorize the mathematical logic of working mathematicians. Where the two men might share knowledge is in the domain of discourse known as mathematical explanation (SEP).

A working mathematician looks for proof and provides mathematical explanation, whereas a philosopher of mathematics seeks to put the practice of mathematical explanation into a philosophical framework to explain how explanation works. The former might be understood as the language game that mathematicians play, and the latter as the language game philosophers play. Mathematicians do math, whereas philosophers are more interested in understanding in a greater context what exactly it is that mathematicians do and how it relates to other topics like science, language, and the mind.

Working mathematicians generally agree on axioms and syntax when exchanging ideas and focus on proof theory, where as philosophers are quite disputatious and are often looking to answer metaphysical, epistemological, or ontological questions that have little to do with mathematical proof. One of the fascinating debates in philosophy of logic and mathematics is whether or not mathematical structure is real or not. Another is whether it is objective and mind-independent. Frege was an antipsychologist and claimed that mathematics and logic were not dependent on the mind. Brouwer was an intuitionist (SEP) who disagreed and sent some working mathematicians down the path of mathematical constructivism as a technique in math. Plato argued there was an alternative realm. Shapiro argues that structures are real and objective. Lakoff argues math is nothing more than neural computation. A good, brief introduction to the philosophy of math is Philosophy of Mathematics (GB) by Linnebo.

So, does math care? Mathematics is a theory, and doesn't care, but the people who do it can roughly divided into two groups: those who don't, the working mathematicians who tend towards Platonism and realism based on their intuitions, and those who do care, mathematical philosophers who have a bewildering array of views about mathematical existence of entities. In this way, formal scientists are a lot like physical scientists. They often see such philosophical disputes as irrelevant and don't spend much time on the question. They simply declare in their papers and books things exist and move on.

You ask:

Does math admit(care about) the existence of an entity?

If you consider math a formal science, and the question of existence an ontological and therefore philosophical question, then its best to say that contemporary, working mathematicians tend not to care so much about what it means for abstract objects (SEP) to exist, and leave such questions to philosophers of mathematics, preferring instead to perform the existential quantification with symbols and leave the truth of the matter intuition (SEP). Historically, there was little distinction between philosophy and mathematics with the ancient Greeks like Pythagoras and Plato being both philosophers and mathematicians. But today, working mathematicians are interested in the science of math, whereas philosophers, particularly philosophers of mathematics, are more interested in the philosophical question.

Andrew Wiles is an example of a working mathematician, and Stewart Shapiro is an example of a philosopher of mathematics. Wiles is famous for his solution to Fermat's Last Theorem, and immersed himself for decades in the mathematical and logical sciences to find a proof related to the conjecture. On the other hand, Shapiro is famous for his philosophical theory which is about various questions and is called structuralism. Wiles dealt with language and formal systems that were distinctly mathematical and logical in character, whereas Shapiro's work deals with natural language that seeks to describe and explain and categorize the mathematical logic of working mathematicians. Where the two men might share knowledge is in the domain of discourse known as mathematical explanation (SEP).

A working mathematician looks for proof and provides mathematical explanation, whereas a philosopher of mathematics seeks to put the practice of mathematical explanation into a philosophical framework to explain how explanation works. The former might be understood as the language game that mathematicians play, and the latter as the language game philosophers play. Mathematicians do math, whereas philosophers are more interested in understanding in a greater context what exactly it is that mathematicians do and how it relates to other topics like science, language, and the mind.

Working mathematicians generally agree on axioms and syntax when exchanging ideas and focus on proof theory, where as philosophers are quite disputatious and are often looking to answer metaphysical, epistemological, or ontological questions that have little to do with mathematical proof. One of the fascinating debates in philosophy of logic and mathematics is whether or not mathematical structure is real or not. Another is whether it is objective and mind-independent. Frege was an antipsychologist and claimed that mathematics and logic were not dependent on the mind. Brouwer was an intuitionist (SEP) who disagreed and sent some working mathematicians down the path of mathematical constructivism as a technique in math. Plato argued there was an alternative realm. Shapiro argues that structures are real and objective. Lakoff argues math is nothing more than neural computation. A good, brief introduction to the philosophy of math is Philosophy of Mathematics (GB) by Linnebo.

So, does math care? Mathematics is a theory, and doesn't care, but the people who do it can roughly divided into two groups: those who don't, the working mathematicians who tend towards Platonism and realism based on their intuitions, and those who do care, mathematical philosophers who have a bewildering array of views about mathematical existence of entities. In this way, formal scientists are a lot like physical scientists. They often see such philosophical disputes as irrelevant and don't spend much time on the question. They simply declare in their papers and books things exist and move on.

Source Link
J D
J D
  • 44.2k
  • 4
  • 35
  • 143

You ask:

Does math admit(care about) the existence of an entity?

If you consider math a formal science, and the question of existence an ontological and therefore philosophical question, then its best to say that contemporary, working mathematicians tend not to care so much about what it means for abstract objects (SEP) to exist, and leave such questions to philosophers of mathematics, preferring instead to perform the existential quantification with symbols and leave the truth of the matter intuition. Historically, there was little distinction between philosophy and mathematics with the ancient Greeks like Pythagoras and Plato being both philosophers and mathematicians. But today, working mathematicians are interested in the science of math, whereas philosophers, particularly philosophers of mathematics, are more interested in the philosophical question.

Andrew Wiles is an example of a working mathematician, and Stewart Shapiro is an example of a philosopher of mathematics. Wiles is famous for his solution to Fermat's Last Theorem, and immersed himself for decades in the mathematical and logical sciences to find a proof related to the conjecture. On the other hand, Shapiro is famous for his philosophical theory which is about various questions and is called structuralism. Wiles dealt with language and formal systems that were distinctly mathematical and logical in character, whereas Shapiro's work deals with natural language that seeks to describe and explain and categorize the mathematical logic of working mathematicians. Where the two men might share knowledge is in the domain of discourse known as mathematical explanation (SEP).

A working mathematician looks for proof and provides mathematical explanation, whereas a philosopher of mathematics seeks to put the practice of mathematical explanation into a philosophical framework to explain how explanation works. The former might be understood as the language game that mathematicians play, and the latter as the language game philosophers play. Mathematicians do math, whereas philosophers are more interested in understanding in a greater context what exactly it is that mathematicians do and how it relates to other topics like science, language, and the mind.

Working mathematicians generally agree on axioms and syntax when exchanging ideas and focus on proof theory, where as philosophers are quite disputatious and are often looking to answer metaphysical, epistemological, or ontological questions that have little to do with mathematical proof. One of the fascinating debates in philosophy of logic and mathematics is whether or not mathematical structure is real or not. Another is whether it is objective and mind-independent. Frege was an antipsychologist and claimed that mathematics and logic were not dependent on the mind. Brouwer was an intuitionist (SEP) who disagreed and sent some working mathematicians down the path of mathematical constructivism as a technique in math. Plato argued there was an alternative realm. Shapiro argues that structures are real and objective. Lakoff argues math is nothing more than neural computation. A good, brief introduction to the philosophy of math is Philosophy of Mathematics (GB) by Linnebo.

So, does math care? Mathematics is a theory, and doesn't care, but the people who do it can roughly divided into two groups: those who don't, the working mathematicians who tend towards Platonism and realism based on their intuitions, and those who do care, mathematical philosophers who have a bewildering array of views about mathematical existence of entities. In this way, formal scientists are a lot like physical scientists. They often see such philosophical disputes as irrelevant and don't spend much time on the question. They simply declare in their papers and books things exist and move on.