From: lexfridman
The interplay between mathematics and language is a fascinating domain that highlights the connection between abstract thought and linguistic expression. Mathematics, often considered the language of the universe, shares significant conceptual and structural parallels with human language. This article explores these parallels, drawing insights from a conversation with mathematician Jordan Ellenberg.
Mathematics as a Form of Language
Mathematics can be seen as a form of language, in that it is a medium through which we express concepts, solve problems, and communicate ideas. Jordan Ellenberg reflects on this, suggesting that the production of mathematical output feels akin to generating linguistic output:
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The feeling of producing mathematical output, if you want, is like the process of, you know, uttering language or producing linguistic output. I think it feels something like that, and it’s certainly the case—let me put it this way—it’s hard to imagine doing mathematics in a completely non-linguistic way. [00:01:58]
Ellenberg notes that even when mathematics involves non-verbal elements like visual proofs, there is a linguistic component involved in reasoning and communication.
Visual Mathematics and Language
There are distinct instances where mathematics transcends verbal language, particularly through visual representations. Ellenberg points out the role of dissection proofs in geometry:
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One thing I talk about in the book is dissection proofs, these very beautiful proofs of geometric propositions. [00:02:40]
These proofs do not necessarily rely on language but on visual understanding, suggesting that mathematics can exist outside of language, yet still communicate ideas effectively through visual means.
Symmetry and Transformation
Symmetry, a concept deeply rooted in both mathematics and language, serves as a bridge connecting the two. Symmetry involves transformation—an inherent characteristic of both mathematical concepts and linguistic constructs:
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In math, we could use symmetry to refer to any kind of transformation of an image or a space or an object. [00:10:42]
In the context of language, transformations occur in grammar and phonetics when creating new forms or meanings from existing words or sentences.
The Process of Language and Math
Mathematics and language both involve a process of manipulating elements to derive meaning or solutions. Ellenberg suggests that mathematics, like language, is a process of aggregation and synthesis:
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Language is a process, and math is a process… It’s in action, it’s ultimately created through action, through change. [00:04:08]
This reflective process is seen in language through syntax and semantics, where rules and structures are applied to communicate effectively.
The Philosophical Perspective
The philosophical inquiry into the nature of mathematics and language raises intriguing questions about their foundations and interconnections. Ellenberg, invoking Noam Chomsky’s theory of universal grammar, questions whether mathematical thinking is as fundamental or more fundamental than language:
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Is it as fundamental as language in the Chomsky view? Is it more fundamental than language? [00:01:37]
The implication here is whether mathematical structures are echoes of the same kind of abstract framework considered by Chomsky in linguistic theory.
Conclusion
The relationship between mathematics and language is intricate and multifaceted, involving vision, abstraction, and transformation. While mathematics often extends beyond the realm of verbal language, the linguistic elements inherent in mathematical reasoning highlight the intertwined nature of these two fundamental human constructs.
Mathematics as a language offers a unique way to communicate abstract ideas, thus serving as a vital tool in both scientific exploration and philosophical reflection. This connection enriches our understanding of both fields and underscores the beauty and complexity of human cognition.
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