I do think there is some difference in kind, not centered in language but stemming from the fact that the basis for validity in mathematics is logic rather than evidence/data, as well as the expectation that mathematics has complete logical fidelity. Logical fidelity is part of what makes mathematics so useful for scientific application. The language becomes a hybrid of human language and (what we can now think of as) computer code.
Great article. I'm into learning languages, and I often look at math as similar to learning a language.
Both involve building vocabulary, mastering grammar-like rules, and gaining fluency through sufficient practice. Symbols and notation act as a dense "regimented ordinary language," allowing compact expression of complex ideas, much like mastering idioms or syntax in French or Spanish. Language has grammar rules fused with cultural nuance, and you abstract from concrete speech. That's similar to math's tight fusion of concepts and procedures, plus its abstract nature.
However, math has stricter progressions vs learning a language. For instance, learning German lets you begin with simple nominative-case sentences like “Der Mann isst Brot” right away and start communicating. Truly internalizing the full case system—accusative, dative, and genitive—requires sufficient repetitive practice (like in math) with declensions, articles, and adjective endings until they become automatic through drills and other practice. But you don't have to master accusative before dative or genitive, or dative or genitive before accusative and you can debate where to start after nominative regardless. There are also rules on adjectives to learn, along with tons of vocabulary and learning to speak it with automaticity.
Elementary math demands stricter sequencing: students must master addition with regrouping, subtraction, place value, multiplication, and long division through sufficient practice until those procedures are fluent and automatic, because they form the non-negotiable procedural and conceptual foundation for fractions, ratios, proportions, and algebra; a weak grasp at any step creates lasting roadblocks.
Thanks Todd. The language analogy is a good one. You want kids to see a sequence of steps in solving an equation to be a sequence of sentences: if 3x = 18 then that must mean that 3x/3 = 18/3 because if two numbers are equal then their quotients by 3 are equal . . . ." The thing that's different about mathematics is the many layers of compression in meaning. It's as if you had a language that would summarize an entire sentence or paragraph with a single word. Amusingly, German, the language you mentioned, kind of does that with its ridiculously long words. But that's really concatenation, not compression. There's a book by Susan Carey, the Origin of Concepts, which I've been meaning to read for years and which describes this compression step as a sort of discontinuity. Maybe I'll read it and write about it some time.
Not my original idea. :) There's a book on my reading list too - Through the Language Glass: Why the World Looks Different in Other Languages. It's about how certain languages shape ways of looking at the world. For instance, with German, they say it really makes you pay attention to what someone is saying because the last word can completely change the meaning of the sentence.
I look at math as the language of precision. For instance, in English, “I saw the man with the telescope” can mean two entirely different things depending on context; in Spanish, “Lo vi con el telescopio” carries similar ambiguity.
There is no ambiguity in math: “If 3x = 18, then 3x/3 = 18/3” means exactly one thing because the equals sign and the division property of equality are defined with zero tolerance for interpretation. Scientific notation offers another striking example — 6.022 × 10²³ instantly and unambiguously conveys Avogadro’s number. And on your compression point, scientific notation here is compressing a 24-digit quantity into a compact, universal form that would be impossible to handle precisely in ordinary language.
Each symbol or step compresses an entire logical paragraph into an unambiguous statement. Unlike natural languages—where words can shift meaning by tone, culture, or omission—math’s layered compression demands and delivers absolute clarity. A single misplaced parenthesis, exponent, or undefined term breaks the logic entirely.
I do think there is some difference in kind, not centered in language but stemming from the fact that the basis for validity in mathematics is logic rather than evidence/data, as well as the expectation that mathematics has complete logical fidelity. Logical fidelity is part of what makes mathematics so useful for scientific application. The language becomes a hybrid of human language and (what we can now think of as) computer code.
Thanks Dev, I kinda left out that whole logic thing, didn't I?
Great article. I'm into learning languages, and I often look at math as similar to learning a language.
Both involve building vocabulary, mastering grammar-like rules, and gaining fluency through sufficient practice. Symbols and notation act as a dense "regimented ordinary language," allowing compact expression of complex ideas, much like mastering idioms or syntax in French or Spanish. Language has grammar rules fused with cultural nuance, and you abstract from concrete speech. That's similar to math's tight fusion of concepts and procedures, plus its abstract nature.
However, math has stricter progressions vs learning a language. For instance, learning German lets you begin with simple nominative-case sentences like “Der Mann isst Brot” right away and start communicating. Truly internalizing the full case system—accusative, dative, and genitive—requires sufficient repetitive practice (like in math) with declensions, articles, and adjective endings until they become automatic through drills and other practice. But you don't have to master accusative before dative or genitive, or dative or genitive before accusative and you can debate where to start after nominative regardless. There are also rules on adjectives to learn, along with tons of vocabulary and learning to speak it with automaticity.
Elementary math demands stricter sequencing: students must master addition with regrouping, subtraction, place value, multiplication, and long division through sufficient practice until those procedures are fluent and automatic, because they form the non-negotiable procedural and conceptual foundation for fractions, ratios, proportions, and algebra; a weak grasp at any step creates lasting roadblocks.
Thanks Todd. The language analogy is a good one. You want kids to see a sequence of steps in solving an equation to be a sequence of sentences: if 3x = 18 then that must mean that 3x/3 = 18/3 because if two numbers are equal then their quotients by 3 are equal . . . ." The thing that's different about mathematics is the many layers of compression in meaning. It's as if you had a language that would summarize an entire sentence or paragraph with a single word. Amusingly, German, the language you mentioned, kind of does that with its ridiculously long words. But that's really concatenation, not compression. There's a book by Susan Carey, the Origin of Concepts, which I've been meaning to read for years and which describes this compression step as a sort of discontinuity. Maybe I'll read it and write about it some time.
Not my original idea. :) There's a book on my reading list too - Through the Language Glass: Why the World Looks Different in Other Languages. It's about how certain languages shape ways of looking at the world. For instance, with German, they say it really makes you pay attention to what someone is saying because the last word can completely change the meaning of the sentence.
I look at math as the language of precision. For instance, in English, “I saw the man with the telescope” can mean two entirely different things depending on context; in Spanish, “Lo vi con el telescopio” carries similar ambiguity.
There is no ambiguity in math: “If 3x = 18, then 3x/3 = 18/3” means exactly one thing because the equals sign and the division property of equality are defined with zero tolerance for interpretation. Scientific notation offers another striking example — 6.022 × 10²³ instantly and unambiguously conveys Avogadro’s number. And on your compression point, scientific notation here is compressing a 24-digit quantity into a compact, universal form that would be impossible to handle precisely in ordinary language.
Each symbol or step compresses an entire logical paragraph into an unambiguous statement. Unlike natural languages—where words can shift meaning by tone, culture, or omission—math’s layered compression demands and delivers absolute clarity. A single misplaced parenthesis, exponent, or undefined term breaks the logic entirely.