HOUSE_OVERSIGHT_015889.jpg
1.26 MB
Extraction Summary
3
People
1
Organizations
0
Locations
0
Events
1
Relationships
2
Quotes
Document Information
Type:
Book manuscript page / evidentiary document
File Size:
1.26 MB
Summary
This document is page 199 of a manuscript titled 'Known Unknowns', bearing a House Oversight evidence stamp. The text discusses the definition of numbers and the Peano Axioms, utilizing a dialogue between a child and their mother ('Mummy') to explain that mathematical symbols are arbitrary definitions. This appears to be part of a personal writing or book draft seized during an investigation.
People (3)
| Name | Role | Context |
|---|---|---|
| Giuseppe Peano | Mathematician |
Mentioned as the creator of the Peano axioms for numbers.
|
| Mummy | Mother of the narrator |
Participant in a dialogue explaining axioms to the narrator.
|
| Mr. Peano | Mathematician |
Referenced in the dialogue as the authority on why numbers are defined as they are.
|
Organizations (1)
| Name | Type | Context |
|---|---|---|
| House Oversight Committee |
Implied by the footer 'HOUSE_OVERSIGHT_015889', indicating this is a seized document used in a congressional investig...
|
Relationships (1)
Dialogue addressing 'Mummy' regarding math questions.
Key Quotes (2)
""Why is the lowest number zero, Mummy?" "Because I say so!" Or, at least "...because Mr. Peano said so." That's what an axiom is."Source
HOUSE_OVERSIGHT_015889.jpg
Quote #1
"It helps us to learn the numbers because 1 is a single line, 2 is two lines joined, three is basically three lines looped together, and four is four lines"Source
HOUSE_OVERSIGHT_015889.jpg
Quote #2
Full Extracted Text
Complete text extracted from the document (1,942 characters)
Known Unknowns
199
199
Numbers
It was not until the late 18th century that numbers were properly codified.
The mathematician Giuseppe Peano gave us the rules, so they are called
Peano axioms. Here are his 'axioms' in natural language.
It was not until the late 18th century that numbers were properly codified.
The mathematician Giuseppe Peano gave us the rules, so they are called
Peano axioms. Here are his 'axioms' in natural language.
Peano Axioms
1. The first number is named zero.
2. Every number has a next number (called its successor). Example:
the next number after one is two.
3. Numbers are singular. Every number with the same name is the
same thing.
4. If something is true of a number, it should be true of the next
number (the successor number).
2. Every number has a next number (called its successor). Example:
the next number after one is two.
3. Numbers are singular. Every number with the same name is the
same thing.
4. If something is true of a number, it should be true of the next
number (the successor number).
From this we can prove some very simple things.
1+1=2. Because the next number after 1 is 2 and '+1' means take
the successor. (You can see I cheated here a little and did not take 100
pages for the proof.)
Back to my poor mother: "Why is the lowest number zero,
Mummy?" "Because I say so!" Or, at least "...because Mr. Peano said
so." That's what an axiom is.
"OK, but why is 3 greater than 2."
"Because I said that each number has a thing that comes after it.
"But, why can't 3 come after zero!"
"It can!"
"But then, if 3 is the thing after zero, I could count 0, 3, 2, 4..."
"Yes, if you want to..."
"I'm sort of lost. Now, you are saying that 3 doesn't really 'mean'
anything. It just comes after 0."
"Yes. You can make up any symbols you like. You just have to
remember what you said and be consistent."
The dialogue shows the importance of definition in mathematics.
I could define my counting numbers as 0, 1, 2, 3, 4 or as o, π, ρ, σ, ς, or
亨, 仇, 仕, 仝 or to be really annoying and confusing 0, 3, 1, 2, 4; they
are only arbitrary symbols. It helps us to learn the numbers because 1
is a single line, 2 is two lines joined, three is basically three lines looped
together, and four is four lines, but we could have used any symbols
we cared for. It is the rules for manipulating these symbols that are the
important part and give mathematics its meaning.
1+1=2. Because the next number after 1 is 2 and '+1' means take
the successor. (You can see I cheated here a little and did not take 100
pages for the proof.)
Back to my poor mother: "Why is the lowest number zero,
Mummy?" "Because I say so!" Or, at least "...because Mr. Peano said
so." That's what an axiom is.
"OK, but why is 3 greater than 2."
"Because I said that each number has a thing that comes after it.
"But, why can't 3 come after zero!"
"It can!"
"But then, if 3 is the thing after zero, I could count 0, 3, 2, 4..."
"Yes, if you want to..."
"I'm sort of lost. Now, you are saying that 3 doesn't really 'mean'
anything. It just comes after 0."
"Yes. You can make up any symbols you like. You just have to
remember what you said and be consistent."
The dialogue shows the importance of definition in mathematics.
I could define my counting numbers as 0, 1, 2, 3, 4 or as o, π, ρ, σ, ς, or
亨, 仇, 仕, 仝 or to be really annoying and confusing 0, 3, 1, 2, 4; they
are only arbitrary symbols. It helps us to learn the numbers because 1
is a single line, 2 is two lines joined, three is basically three lines looped
together, and four is four lines, but we could have used any symbols
we cared for. It is the rules for manipulating these symbols that are the
important part and give mathematics its meaning.
HOUSE_OVERSIGHT_015889
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