'Natural Numbers' - Words used to count. Created from Abstract Categories or Groups. "Plural" - Combination of Numbers/Words - Small Numbers different function to Large Numbers (logically reach up to 6/7 before start to count).
-(8545)- Basic Symbols Organised According to Syntax (inferred system of rules). Can be Analysed Through Division, etc. Addition and Multiplication as "Plurals of Plurals". Technology as innate part of Neurological Syntax. [CHOMSKY] A Basic Logical Language containing Limited Human Intelligence (predictive text, etc).
Three attitudes towards numbers:
1) They are natural and empirical/observed. [RUSSELL]
2) They are intuitive [PYTHAGORAS/PLATO] - Numbers are from a different world. "Supernatural".
3) They are abstract objects [FREGE] that modify meaning. Not something we learn so must be innate [CHOMSKY]
Apes able to distinguish Simple Plurality. 0 (the absence of a thing/"one thing"), 1 (one/enough/"more than one thing") and 2 (a lot/unlimited/more than one/"many things") - All are strong natural numbers.
FREGE and "The Foundations of Arithmetic" [1953] Against PLATO - Mathematics and Arithmetic Logical. 0 not found in nature, therefore must be derived from logic; Apriori. "Axiom" - All things are identical/equal. Pairs identical to other pairs (2 peaches, 2 oranges, etc). Larger numbers as "pairs of pairs", we merely attached the symbol '4', for example.
FREGE'S METHOD. Numbers as 'Logical Objects'. Every number has its own equivalent. Zero the class of objects that are not equal. Contradicts RUSSELL - Numbers Observed - Concept of "Zero" not exist by Definition. Rejection of Empiricism. Zero not found in Nature and +1 not a constant (natural number must have successor - negative numbers not used for counting), therefore all numbers derive from logic.
PYTHAGOREANISM. Prime Numbers Pre-Existing/"supernatural" - Necessary condition for 'Consciousness'. Religious significance of the number 3 (First Plural Prime); Holy Trinity, 3 Crosses, Resurrected 3 days. FRAZER and "The Golden Bough"/"Apocalypse Now" - Based around Plural Prime 1. All Numbers Rational Combinations of Prime ("One" and "Not One") They Exist in a State of 'Non-Dimension' - PLATO's "Perfect" World.
NOTHING AND ZERO.
-(Zero = Something = Nothing)-
ARISTOTLE'S 'Law of Contradiction'. Solved by LEIBNIZ "Monads" - A Single Object can contain own Essence. "Nothing" as Philosophical Absurdity. HEIDEGGER - Impossible due to Qualitative Gap between "Nothing" (0) and "Something" (1).
RUSSELL. "Principia Mathematica" (1913) Numbers have Objective Existence - "The Whole is Bigger Than The Sum of The Parts" - Understand Individual Element by Understanding the System as a Whole. Each Proposition must be Verified. [POPPER].
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"FREGE on Logic, Psychology and Epistemology" (page 155)
"Begriffsschrift" [Translation "Concept Writing", 1879] - Relationship Between Epistemology (knowledge) and Language. Proposition before Proof - must be Justification. Mathematical and Apriori. Arithmetic consists of different images, same number. Deals with Truth of the Proposition.
"A proposition may be thought of without being true, and a proposition may be true without being thought of" (page 156)
For example, a sentence can make sense without reference to external - SYNTAX (order of words) [RUSSELL].
FREGE'S EGO. Consciousness as Object of Thought. Anti-CARTESIAN. Experience/Aposteriori. "We do not have thoughts, thoughts are what we grasp" (page 159) - Thoughts do not change, it's Action that changes our thoughts. Public/Physical and Private/Consciousness + The 'Third Realm' (the world of thoughts).
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"Knowledge by Acquaintance and Knowledge by Description" (page 160)
BERTRAND RUSSELL and the "Problems of Philosophy" [1912]. Mathematics as Empirical Science. External World is Instinctive - No proof that life isn't just a dream [CARTESIAN DOUBT].
'Sense Data' - that of which we are Certain. Private and Personal. This is described as 'Knowledge by Acquaintance' - internal thought. 'Knowledge by Description' describes the physical objects that cause this 'Sense Data'.
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"WITTGENSTEIN on Certainty" (page 165)
The 'Language-Game'. "p one must understand what is mean't by saying p" (page 165) Private Language - Consciousness with reference to External. We question the meaning of the words. Some Propositions cannot be doubted as it's possible to dream that words have meaning [DESCARTES].
"On Certainty" [1969] These Propositions are not learned from experience. They are based on the structure of our own knowledge. Our actions determine the result.
"The system is something that a human being acquires by means of observation and instruction" (page 167)
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"A Guide to Understanding Nothing" [Honan, D. 06/02/13]
Idea that the universe was created out of nothing. Knowledge gained from observation/experience. "A Universe from Nothing: Why There is Something Rather Than Nothing" [LAWRENCE KRAUSS, 2012] Quantum Mechanics and Relativity [EINSTEIN] - Particles that flash in and out of existence.
"Empty space with stuff in it"
"Empty space with nothing in it" - as same.
Nothing will eventually produce something, though according to 'law of physics' and 'law of nature', this is not clear or necessary. "The laws themselves come into existence when the universe comes into existence" - There is no pre-existing fundamental law.
No laws, no space, no time, no particles, no radiation = NOTHING.
Sources:
Honan, Daniel "A Guide to Understanding Nothing" (2013)
Kenny, Anthony "Philosophy in the Modern World" (2007)
the title does not go with the content! refs a different topic!
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