Group:
information
Group:
sets
Topic:
abstraction
Topic:
abstraction in programming
Topic:
analytic truth
Topic:
discrete vs. continuous
Topic:
elements
Topic:
reality is a machine
Topic:
history of mathematics
Topic:
infinity and infinitesimal
Topic:
kinds of numbers
Topic:
mathematics as a formal system
Topic:
natural language as a system
Topic:
number and arithmetic as part of language
Topic:
number as a named set of numbers
Topic:
number as a progression for counting and 1-1 relations
Topic:
number as the extension of a class of equinumerous classes
Topic:
number representation
Topic:
state
Topic:
thought is computational
Topic:
unique numeric names as surrogates
Topic:
value as an abstraction
Topic:
what is a computer
Topic:
what is truth
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Summary
Number is an important concept with many practical applications. How ever we understand number, it should match its actual use. In particular, we use numbers for counting, communication and computation.
Numbers are not sets or sets of unit objects. If they were, then they would need to be distinguishable and indistinguishable at the same time. Nor do they belong to a concept. Instead numbers apply to concepts. This explains their wide applicability. We abstract a concept from experience. We then enumerate or count the concept. For example, "Venus has 0 moons" is a statement about the concept "moon of Venus".
Computers have brought a new role to number. They manipulate unitless numbers as state by executing unitless numbers as program. Users encode a problem is some form and then decode the result.
The concepts of set, function, number, computation, and formal logic are tightly intertwined. Despite this, we should not lose sight of the role of number in language. They are used for communication and they act as quantification operators. (cbb 3/94)
Subtopic: number as primary 
| Quote: Pythagoreans saw numbers as the primary natures, the elements of all things; including justice, soul, mind, musical modes, relations, heaven [»aris_322a]
| | Quote: all truths can be explained by numbers [»leibGW_1686]
| | Quote: the drudgery of the numbers will make you free [»geneH_1984]
|
Subtopic: number as innate
| Quote: preschool children can compare and add large sets of visual and auditory elements without counting; addition as accurate as comparison [»bartH9_2005]
| | Quote: abstract knowledge of number and addition precedes, and may guide, language-based instruction in mathematics
| | Quote: animals have a number category and some understanding of a number concept, e.g., ordinality [»hausMD_2000]
| | Quote: with difficulty, animals can discriminate numbers 0 to 10 as well as young children [»hausMD_2000]
| | Quote: different process of acquistion tells us that animals and children represent number in different ways
|
Subtopic: number as language
| Quote: number and language skills leapfrog each other as children learn to manipulate symbols
|
Subtopic: number as concept
| Quote: number applies to concepts that are abstracted from things; this explains number's wide range of applicability [»fregG_1884]
| | Quote: analyze categories as mutipart numbers; e.g. 'Interval' is 2.3.10 where 2 is space, 3 is between, and 10 is whole [»leibGW_1666]
| | Quote: a thing is called one or single simply with respect to its existence; number applies to things with a common genus, e.g., coins [»fregG_1884]
| | Quote: a statement of number is an assertion about a concept; e.g., "Venus has 0 moons" concerns the concept "moon of Venus" [»fregG_1884]
| | Quote: oneness is a characteristic of the higher order concept of all unitary concepts; this differs from genus and species [»fregG_1884]
| | Quote: statements such as "All whales are mammals" and of number concern concepts instead of objects; an indefinite object is really a concept [»fregG_1884]
| | Quote: numbers have an inherent quality to them; they can be accurate or fuzzy, precise or rounded off, detailed or averaged and vague
| | Quote: Ockham's razor--plurality [of reasons] is not to be posited without necessity
| | Note: the primitive concept is the natural numbers, everything else is convention [»cbb_1990, OK]
|
Subtopic: number as answer
| Quote: mathematical entities are eternal, unchanging, and many alike; intermediate between changeable objects and unique ideas [»aris_322a]
| | Quote: every number is the answer to squillons of questions [»fynn_1974]
| | Quote: picking a time for "inhabitant of Germany", fixes the number for all eternity
| | Quote: numbers in business are symbols, like words, with rich meanings from relationships to one another
| | Quote: the key issue in business is finding out what is happening behind the numbers
|
Subtopic: number as information
| Note: information is numbers, what everyone can agree on; the problem is a vivid language for numbers [»cbb_1990, OK]
|
Subtopic: number as count
| Quote: use tally sticks or parts of your body to keep track of numbers; no need to count as we do [»ifraG_2000]
| | Quote: beyond four, quantities are vague; must count to find out [»ifraG_2000]
| | Quote: use twin tally sticks to record sums owed and settled; marked together with a saw [»ifraG_2000]
|
Subtopic: number as mathematics
| Quote: the sole natural object of mathematical thought is the whole number [»poinH_1905, OK]
|
Subtopic: number as things
| Quote: mathematical entities are not separate from sensible things; not first principles; no coherency [»aris_322a]
| | Quote: the concept of number should apply in the right way to common objects
| | Quote: identity statements are senseless outside of a consistent context for individuating things [»benaP_1965]
|
Subtopic: number as neither things nor concepts
| Quote: number can not be defined as a collection of unit objects; if distinguished, can't do arithmetic; if not, they merge together [»fregG_1884]
| | Quote: can not define numbers as belonging to a concept; e.g., is Caesar a number? What is the object 0? [»fregG_1884]
| | Quote: there are many ways to define numbers as sets; each defines relations differently; so number is not a particular set [»benaP_1965]
|
Subtopic: number as set
| Quote: set theory investigates the fundamental notions of number, order, and function [»zermE19_1908]
| | Quote: number as a set of units; even number as equal parts [»eucl_300, OK]
| | Quote: every rational and irrational number is symbol for a cut that divides the real numbers into two
| | Quote: construct the real numbers as segments of series of rational ratios in order of magnitude; an irrational number is a segment without boundary [»russB_1919, OK]
| | Quote: all things involve number and number is that which is composed of units [»alkhMM_825, OK]
|
Subtopic: number as quantifier
| Quote: number words function like quantification operators; if numbers exist then so should quantifiers [»benaP_1965]
|
Subtopic: number as program
| Quote: infinite things can be compounded out of the combination of a few; e.g., numbers from digits [»leibGW_1679]
| | Note: numbers are transition tables of Turing machines and vice versa; numbers and programs are the same [»cbb_1990, OK]
| | Note: a number is what we exactly agree about; what we exactly agree about can be represented by named sets; i.e., Church's thesis [»cbb_1990, OK]
| | Quote: the computable numbers are the real numbers whose decimal expression can be calculable by finite means; because human memory is limited [»turiAM11_1936]
|
Subtopic: number as sequence/induction
| Quote: the natural numbers are the posterity of 0 w.r.t. immediate predecessor; i.e., all larger numbers, by hereditary; includes Peano's 0 and induction [»russB_1919, OK]
| | Quote: we define the natural numbers as those to which proofs by mathematical induction can be applied
| | Quote: the fraction m/n is the relation that holds of inductive numbers x,y when xn=ym; leads to ordering relation on fractions [»russB_1919, OK]
|
Subtopic: number as word sequence
| Quote: India has a place-value system of word numerals; 1 is anything markedly unique, e.g., the moon, the earth; 0 is denoted by words meaning void, sky, complete; still used today [»dattB_1935]
|
Subtopic: number as state
| Quote: a state in a state machine is a unitless number [»cbb_1980, OK]
| | Quote: a data object is a number since it is part of the initial state of a computation
| | Quote: computers only operate on the marks of numbers; humans set up a correspondence between any set of things and numbers [»jakiSL_1969]
|
Subtopic: number as name
| Quote: a Godel number is a name for a string [»hofsDR_1979, OK]
|
Subtopic: number as attribute
| Quote: number is entirely the creature of the mind; 1 house has many windows, 1 city has many houses [»fregG_1884]
| | Quote: what number belongs to a pile of playing cards depends on how we chose to regard it; e.g., as packs [»fregG_1884]
| | Quote: FregeG and RussellB see numbers as logical attributes of a concept; i.e., its cardinality [»carnR_1931]
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Related Topics
Group: information (46 topics, 1160 quotes)
Group: sets (7 topics, 148 quotes)
Topic: abstraction (62 items)
Topic: abstraction in programming (67 items)
Topic: analytic truth (51 items)
Topic: discrete vs. continuous (47 items)
Topic: elements (22 items)
Topic: reality is a machine (48 items)
Topic: history of mathematics (57 items)
Topic: infinity and infinitesimal (37 items)
Topic: kinds of numbers (24 items)
Topic: mathematics as a formal system (30 items)
Topic: natural language as a system (43 items)
Topic: number and arithmetic as part of language (30 items)
Topic: number as a named set of numbers (15 items)
Topic: number as a progression for counting and 1-1 relations (22 items)
Topic: number as the extension of a class of equinumerous classes (23 items)
Topic: number representation (16 items)
Topic: state (35 items)
Topic: thought is computational (60 items)
Topic: unique numeric names as surrogates (67 items)
Topic: value as an abstraction (25 items)
Topic: what is a computer (62 items)
Topic: what is truth
(67 items)
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