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LatineSummary
This lesson opens the fourth unit of the course, devoted to mathematics and logic. It explains what the unit is for: to fill the architecture of the previous unit with formulas, equations, statements and deductions typeset with the precision with which they are thought; it presents its five lessons and the thread that joins them, from mathematical mode to deduction trees; it specifies what is taken for granted from the previous units; it recalls how mathematical and logical notation came into being and why typesetting it was, for centuries, the most expensive work of the printing trade; it states the problem the unit solves, that a formula typed as text looks right and is not; and it shows that the same notation serves physics, economics, computer science and philosophy.
Learning objectives
- Describe the sequence of contents of the unit and explain what each lesson contributes to the next.
- Recognise what the unit takes for granted from the previous ones and review it if necessary.
- Distinguish, with an example, a formula typeset in mathematical mode from one typed as text.
- Recognise in one's own discipline the statements, formulas and deductions that the unit teaches how to typeset.
Entry assessment
Before working through the unit, it is worth measuring the starting point. This assessment is a self-assessment: it is not graded, and its only purpose is to guide your study. Its first two questions review what the unit takes for granted from the previous ones; the next ten cover, two by two and in order, the five lessons of the unit. Anyone who has never written a formula in LaTeX should not expect to get them right: every answer, right or wrong, explains the point and names the lesson that deals with it. The same skills are measured again at the end, in the synthesis lesson.
Before starting the unit, measure your starting point. It is a short self-assessment: it is not graded and its result is stored only in your browser.
Question 1From earlier unitsIn section 2 of an article, whose first numbered equation bears the number 3, the author writes
\label{eq:ohm}on the line before\begin{equation}. What does\eqref{eq:ohm}print?Proximity is not enough: what counts is the order, and a label written before the equation does not see its number.The label exists and is recorded; after the second run it prints a number, though not the intended one.\labelmay be written anywhere in the body; that is why the defect passes without warning.A label records the last number assigned before it, and before the equation the last one was that of the section. Inside the environment, the label would record 3;\eqrefonly adds the parentheses.Review: Cross-references
Question 2From earlier unitsTo write vectors in bold, a preamble contains
\newcommand{\vec}[1]{\mathbf{#1}}. What happens?That is what\renewcommandwould do;\newcommandrefuses to redefine.A name designates a single command.Commands of one's own also work inside formulas; this one, however, is rejected.\newcommanddoes not redefine existing commands. Another name is chosen, such as\vect, or the command is redefined knowingly with\renewcommand, giving up the arrow.Question 3In a document with
\usepackage[english]{babel}and nothing else that touches numbers, what does$12,345$print?babel in English converts neither the point nor the comma: the comma remains punctuation.The comma that is written is printed; what goes wrong is the space after it.The comma is allowed; only its spacing changes.In mathematical mode the comma is punctuation and is followed by a thin space. When the comma belongs to the number, it is written in braces,$12{,}345$, which is correct in any preamble.Review: Inline and displayed formulas
Question 4A sentence ends with a displayed formula. Where does the full stop go?
The full stop would be left alone at the beginning of a line.They are punctuated as if they were inline.The blank line opens a new paragraph, and the full stop would be indented.A displayed formula is part of the sentence; a full stop written after\]would go to the beginning of the next line.Review: Inline and displayed formulas
Question 5What does
$x^10$print?That would needx^{10}: without braces, only the 1 goes up.That error appears with two^on the same base, not with a two-digit exponent.The 0 stays on the baseline, but the 1 does go up as an exponent.The sign^acts on a single element; to group, braces are written:x^{10}.Review: Symbols and notation
Question 6In a document with babel in Spanish and amsmath, what does
$\sin x$print?babel does not redefine\sin; it offers\senseparately.That happens whensin xis written without a backslash.\sinremains defined in the LaTeX kernel.babel in Spanish defines\senas a new command and puts an accent on\lim,\max,\minand\inf, but does not touch\sin.Review: Symbols and notation
Question 7An
alignhas two equations, and the second ends with\\. Which numbers appear in the margin?It ignores it inside\[…\], not in an alignment environment, where it opens a row.That error is produced by an&outside an alignment, not by a double backslash.That happens inalign*, which does not number;alignnumbers the empty line too.\\opens a new line; at the end of the last one it creates an empty line, whichalignnumbers without any warning.Review: Multi-line equations
Question 8A four-line derivation, aligned at its equals signs, must be cited as a whole, with a single number. What should be written?
gatherdoes not align and numbers every line.They are not aligned and carry no number.multlineis for a single formula that is too long, not for an aligned chain.splitaligns the lines and leavesequationa single number, centred with respect to the whole.Review: Multi-line equations
Question 9An article declares
\newtheorem{thm}{Theorem}and\newtheorem{lem}[thm]{Lemma}, and its text contains, in this order, a lemma, a theorem and another lemma. How is the last one labelled?That would be so with separate counters; the argument[thm]makes the lemma share the theorem's counter.Numbering within the section requires[section]at the end of the theorem's declaration.No counter is reset here: without[section], the series runs on.The lemma uses the theorem's counter: the three statements form a single series, and the number tells the position, not the label.Review: Theorems and proofs
Question 10A proof ends with a displayed formula
\[…\], and the final square appears on its own, on an empty line, below it. What corrects this?The environment puts its own square when it closes, and there would be two.The number occupies the right margin, and the square still drops to a line of its own.That changes the shape of the symbol, not its place.When the environment closes, the formula has already been typeset;\qedhereputs the square exactly where it is written.Review: Theorems and proofs
Question 11A logic text wants to write, as a formula of the language it studies, the conditional of p and q. Which source is appropriate?
The double arrow belongs to the metalanguage: it asserts that one thing follows from another.It is the amsmath arrow for linking statements, with wide spacing; it is not the connective either.It asserts semantic consequence; it does not write a formula.The single arrow is the connective of the object language.Review: Logic and deductions
Question 12In bussproofs one writes
\AxiomC{$p$},\AxiomC{$q$},\AxiomC{$r$}and\BinaryInfC{$s$}, and then\DisplayProof. What happens?A binary inference takes only the last two trees.That error appears when trees are missing, not when there are too many.\DisplayProofrequires a single tree on the stack.The binary inference consumes the last two trees, q and r; the one for p is left apart, and\DisplayProofrequires exactly one.Review: Logic and deductions
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