Mathematics is written with symbols, but mathematical meaning lives in relationships among those symbols. Tokenisation supplies the pieces; mathematical parsing and reasoning reconstruct the structure.
An equation such as 3(x + 2) = 15 can be represented as characters, language-model subwords, formal lexer tokens, an expression tree or an algebraic object. Each representation supports different operations.
This article continues the eduKateSingapore Representation and Tokenisation series and connects numerical representation with formal mathematical structure.
The Mathematical Representation Route
MATHEMATICAL IDEA → NOTATION → CHARACTERS / LATEX / MARKUP → TOKENISATION → SYMBOLIC TOKENS → PARSING → EXPRESSION TREE / FORMAL STRUCTURE → TRANSFORMATION / PROOF / COMPUTATION → RENDERED RESULT → VERIFICATION
1. Mathematical Objects Are Not Their Written Symbols
The number two can be written as 2, II, 10₂ or the word two. The underlying quantity remains related while the surface tokenisation changes completely.
Notation is a representation, not the mathematical object itself.
2. Symbols Are Context-Dependent
x can be an unknown, a variable, a coordinate, a multiplication symbol in another typography or simply the letter x in prose.
Mathematical interpretation depends on context and grammar.
3. Operators Are Small Tokens With Large Meaning
+, -, ×, ÷, =, < and ∈ can each change the entire relation expressed by an equation.
Character count has little relationship to conceptual weight.
4. Unary and Binary Operators Can Share the Same Symbol
The minus sign in -5 is unary negation; in 8 - 5 it is binary subtraction.
The same token receives a different structural role depending on position and surrounding expression.
5. Parentheses Create Explicit Grouping
2(3 + 4) and 2×3 + 4 contain similar symbols but produce different structure and value.
Tokenisation preserves parentheses; parsing interprets the hierarchy they create.
6. Operator Precedence Is Not Written Between Every Token
In 2 + 3 × 4, conventional precedence makes multiplication bind before addition.
The surface sequence alone does not list the parse tree explicitly; mathematical grammar supplies it.
7. Associativity Also Changes Structure
Expressions such as a - b - c require a convention for grouping. Left and right associativity can yield different results for some operators.
Formal interpretation depends on syntax rules above the token layer.
8. Fractions Are Two-Dimensional in Print
A typeset fraction places numerator above denominator. Plain text often converts the same structure to a/b. LaTeX may encode it as \frac{a}{b}.
The same mathematical relation has several surface representations with different token sequences.
9. Superscripts Carry Structural Meaning
x², x^2 and x^{2} can all represent exponentiation depending on notation.
A tokenizer can preserve the characters while a parser reconstructs the exponent relation.
10. Subscripts Can Mean Index, Label or Variable Identity
a₁ can indicate the first element of a sequence, while chemical and tensor notation use subscripts differently.
Visual position can be semantic.
11. Two-Dimensional Layout Is Lost in Plain Text
Matrices, aligned equations and piecewise functions use vertical and horizontal arrangement to encode structure.
Flattening them into plain text requires an explicit serialization language or risks losing relationships.
12. LaTeX Is a Serialization Language for Mathematical Structure
LaTeX represents mathematical notation through commands, braces and symbols such as \frac, \sum, _ and ^.
The model tokenizer therefore sees the source language of a typesetting system rather than the rendered equation directly.
13. LaTeX Tokens Are Not Mathematical Tokens Automatically
The string \frac can be one model token or several. Its braces may be separate tokens. Yet mathematically the whole construction represents a fraction operator with numerator and denominator children.
Surface tokenisation and formal syntax remain distinct.
14. Equivalent LaTeX Can Render the Same Mathematics
Whitespace, macro choices and grouping conventions can differ while rendering an equivalent expression.
String equality is therefore stricter than mathematical equivalence.
15. Rendered Equality Is Still Not Algebraic Equality
x + x and 2x look different but are algebraically equivalent under ordinary assumptions.
Mathematical identity can live above both token and rendered form.
16. Expression Trees Capture Structure
An expression tree can represent 2 + 3 × 4 with addition at the root and multiplication as one child.
This makes operator precedence explicit and creates a representation better suited to symbolic manipulation.
17. Prefix and Postfix Notation Encode Structure Differently
The same expression can be written in infix, prefix or postfix notation. Prefix and postfix forms can reduce the need for parentheses because order itself encodes the tree.
Notation redistributes structural information.
18. Polish Notation Shows That Surface Order Is Conventional
A mathematical relation can survive a change in linear order when the grammar changes consistently.
The underlying structure is more fundamental than one notation’s token sequence.
19. Variables Need Scope and Binding
In ∫ f(x) dx, the variable x has a role bound by the integral expression. In logic, quantifiers such as ∀x and ∃x create explicit variable scope.
Symbol identity alone does not determine binding.
20. Bound and Free Variables Are Different Structural States
The same letter can be bound in one subexpression and free in another.
Formal reasoning requires scope-aware representation beyond lexical tokens.
21. Renaming Bound Variables Can Preserve Meaning
Changing a bound variable name from x to y can leave an expression semantically equivalent if all bound occurrences are changed consistently.
This is another case where surface token difference does not imply mathematical difference.
22. Functions Are Higher-Level Units
sin(x), log(x) and f(x) combine a function identity with an argument structure.
A tokenizer may split the characters while a mathematical parser represents one function-application node.
23. Implicit Multiplication Is a Human Convention
2x means multiplication even though no multiplication symbol appears.
The parser must infer an operation from adjacency and notation rules.
24. Juxtaposition Can Be Ambiguous Across Domains
AB might mean multiplication, a line segment, matrix product, concatenation or a named object depending on mathematical domain.
Domain context changes the grammar.
25. Commas Can Separate Coordinates, Arguments or Thousands
(2,3) may be a coordinate pair; f(2,3) may contain two arguments; 2,300 may be a formatted integer.
Punctuation meaning emerges from structure.
26. Set Notation Has Its Own Token Grammar
Braces, membership symbols, unions, intersections and set-builder bars encode relations different from arithmetic operators.
A symbol’s meaning is domain- and context-sensitive.
27. Logic Uses Mathematical Tokens With Formal Semantics
Symbols such as ¬, ∧, ∨, ⇒, ∀ and ∃ participate in strict formal systems.
Formal parsers can evaluate well-formedness independently of natural-language interpretation.
28. Proofs Mix Natural Language and Formal Notation
A mathematical proof can alternate prose, equations, diagrams and symbolic derivations.
Strong reasoning systems need multimodal or multi-representation alignment even inside mathematics.
29. A Proof Step Is a Higher-Level Token of Reasoning
At one granularity, the proof contains characters and symbols. At a higher level, it contains moves such as substitution, contradiction, induction or applying a theorem.
Expert reasoning often operates on these coarse structural units.
30. Symbolic Algebra Systems Use Canonical Internal Forms
Computer algebra systems parse input into structured expressions and transform those structures under algebraic rules.
They do not rely on raw text tokenisation to decide equivalence.
31. Canonicalisation Can Simplify Equality Tests
Expressions can be simplified or normalized into forms that make comparison easier.
But canonical forms depend on domain, assumptions and simplification rules.
32. Assumptions Change Algebraic Validity
√(x²) = x is not universally valid over real x without conditions; it is generally |x|.
Formal transformations need domain assumptions, not only syntactic patterns.
33. Units Add Dimensional Structure
5 m + 3 s is syntactically readable but dimensionally invalid as an ordinary physical sum.
Dimensional analysis adds a semantic type system above mathematical syntax.
34. Units Can Be Parsed Like Algebraic Expressions
kg·m/s² has multiplicative, divisive and exponent structure.
35. Tables and Graphs Are Mathematical Representations Too
A function can be represented as an equation, table or graph. The mathematical object is related across all three forms while the token structures differ radically.
Learning often depends on moving correctly between representations.
36. Graphical Mathematics Needs Visual Tokenisation
A plotted curve, geometry diagram or number line contains spatial relationships that plain-text tokenisation can lose.
Vision and mathematical parsing must work together for image-based mathematics.
37. OCR of Mathematics Is Harder Than OCR of Prose
Superscripts, subscripts, fraction bars, matrices and nested radicals require two-dimensional layout recognition.
Recognising every character is not enough if spatial structure is lost.
38. Math OCR Should Produce Structure, Not Only a String
A high-fidelity pipeline can convert an equation image into LaTeX or MathML that preserves nested relationships.
The target representation should support later parsing and verification.
39. MathML Makes Structure More Explicit Than Visual LaTeX Alone
Mathematical markup languages can encode presentation and semantic relationships in structured trees.
This gives accessibility tools and software a machine-readable route through notation.
40. Accessibility Requires Alternative Mathematical Representations
Screen readers cannot rely on visual placement alone. Structured mathematical markup can verbalise fractions, powers and matrices in a logical order.
Representation must match the receiver’s sensory interface.
41. Language Models Can Learn Mathematical Patterns From Text Tokens
A model can learn many algebraic and symbolic relationships even when its tokenizer was not specifically designed as a mathematical lexer.
Contextual layers reconstruct structure across the subword sequence.
42. But Formal Tools Provide Stronger Guarantees
Symbolic algebra systems, theorem provers and calculators operate under explicit formal semantics. They are better suited to exact manipulation and verification than unconstrained text generation alone.
Strong systems route formal work to formal representations.
43. Tool Calls Require a Representation Bridge
A language model must translate a user’s natural-language request into formal expressions or tool arguments, then translate formal results back into explanation.
The bridge must preserve the mathematical invariants the tool expects.
44. Exact Reconstruction Matters for Mathematical Source
Changing one parenthesis, exponent or minus sign can alter the expression. Generated mathematical notation should therefore be parsed or evaluated when correctness matters.
Visual plausibility is not enough.
45. Equivalent Answers Need Structural Comparison
A student may answer 2(x+1) while the reference answer is 2x+2. String comparison marks them different; algebraic comparison recognises equivalence.
Assessment should compare at the right representation layer.
46. Mathematical Learning Is Granularity Training
Beginners often see isolated symbols. Experts see factors, functions, identities, invariants and proof moves.
Expertise changes what counts as one workable mathematical unit.
47. Misconceptions Can Be Parse Errors
A student may know every symbol but group the expression incorrectly. For example, treating 1/x+1 as 1/(x+1) is a structural misunderstanding rather than vocabulary failure.
Teaching must diagnose the representation layer where the mistake occurs.
48. Representation Switching Can Reveal Invariants
Moving between equation, table, graph and verbal description helps learners identify what remains true while the form changes.
This is the educational power of mathematical representation.
49. The Mathematical Tokenisation Audit
- What mathematical domain applies?
- What notation or markup represents the source?
- Are Unicode symbols, ASCII approximations or LaTeX used?
- How does the language-model tokenizer split the expression?
- What parser reconstructs operator precedence and grouping?
- How are superscripts, subscripts and fractions represented?
- Are variable scope and binding preserved?
- Are units and dimensions represented explicitly?
- Are assumptions attached to transformations?
- Can equivalent expressions be compared structurally?
- Can image-based mathematics preserve two-dimensional layout?
- Is accessible structured markup available?
- Should exact operations route to calculators, CAS or theorem provers?
- Can the final result be verified independently?
50. What Students Should Remember
- Mathematical symbols are representations, not the objects themselves.
- Tokenisation provides pieces; parsing builds expression structure.
- Operator precedence, grouping and scope live above the token layer.
- LaTeX serializes mathematical notation but is not itself the mathematics.
- Equivalent expressions can have different strings.
- Graphs, tables and equations are different representations of related structure.
- Formal tools provide stronger guarantees for exact computation.
51. The Deep Principle
Mathematics works because symbols can be manipulated while preserving invariants. Tokenisation makes the symbols addressable, but mathematics begins when the system recognises which relationships remain true under transformation.
The token tells us what symbol is present. The parse tells us what the symbol is doing. The mathematics tells us what remains true when the representation changes.