Irregularity - Part III

# Name the function

While we wait to start: what would be a suitable name for the function below?

```cpp
template<typename T>
T fn(T a, T b) {
  while (true) {
    if ( b < a ) a -= b;
    else if (a < b) b -= a;
    else return a;
  }
}
```

---

Mircea Baja - 12 June 2020

# Irregularity in Generic Programming

## Part III: Bibliography

---
class: large-points

# Agenda

- Brief history of mathematics - a lot of Euclid and a bit of Gödel
- Stepanov's Elements of Programming
- Brief history of support for concepts in C++

---

# Books

---

# Euclid's Elements

---

# Triplets

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  <text class="svg20181107-01-t1" x="5.0" y="20.0">3</text>
  <text class="svg20181107-01-t1" x="30.0" y="40.0">4</text>
  <text class="svg20181107-01-t1" x="35.0" y="15.0">5</text>
</svg>
</div>

---

# Proof

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      text-anchor: end;
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      font-family: serif;
      font-size: 4px;
      text-anchor: start;
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  <path class="svg20181107-03-l2" d="M 10.0 80.0 L 50.0 50.0 L 50.0 80.0 Z"/>
  <path class="svg20181107-03-l2" d="M 10.0 50.0 L 50.0 50.0 L 10.0 80.0 Z"/>
  <path class="svg20181107-03-l1" d="M 10.0 10.0 L 80.0 10.0 L 80.0 80.0 L 10.0 80.0 Z"/>
  <text class="svg20181107-03-t0" x="65.0" y="55.0">a</text>
  <text class="svg20181107-03-t0" x="42.5" y="30.0">b</text>
  <text class="svg20181107-03-t0" x="67.5" y="30.0">c</text>
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  <path class="svg20181107-03-f2" d="M 140.0 10.0 L 170.0 50.0 L 140.0 50.0 Z"/>
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  <path class="svg20181107-03-l2" d="M 100.0 50.0 L 140.0 50.0 L 100.0 80.0 Z"/>
  <path class="svg20181107-03-l1" d="M 100.0 10.0 L 170.0 10.0 L 170.0 80.0 L 100.0 80.0 Z"/>
  <text class="svg20181107-03-t1" x="122.0" y="30.0">b</text>
  <text class="svg20181107-03-t2" x="122.0" y="30.0">2</text>
  <text class="svg20181107-03-t1" x="157.0" y="65.0">a</text>
  <text class="svg20181107-03-t2" x="157.0" y="65.0">2</text>
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  <path class="svg20181107-03-f2" d="M 220.0 80.0 L 190.0 80.0 L 190.0 40.0 Z"/>
  <path class="svg20181107-03-f2" d="M 190.0 40.0 L 190.0 10.0 L 230.0 10.0 Z"/>
  <path class="svg20181107-03-l2" d="M 230.0 10.0 L 260.0 50.0 L 220.0 80.0 L 190.0 40.0 Z"/>
  <path class="svg20181107-03-l1" d="M 190.0 10.0 L 260.0 10.0 L 260.0 80.0 L 190.0 80.0 Z"/>
  <text class="svg20181107-03-t1" x="227.0" y="45.0">c</text>
  <text class="svg20181107-03-t2" x="227.0" y="45.0">2</text>
</svg>
</div>

---

# Infinity

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      dominant-baseline: middle;
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  <text class="svg20181107-02-t1" x="5.0" y="20.0">1</text>
  <text class="svg20181107-02-t1" x="25.0" y="40.0">1</text>
  <text class="svg20181107-02-t2" x="30.0" y="15.0">sqrt(2)</text>
</svg>
</div>

---

---
class: large-points

# Definitions

- B.I D.1: A point is that of which there is no part.
- ...
- B.I D.22: And of the quadrilateral figures: a square is that which is
  right-angled and equilateral ...

---

# Postulates

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  <path class="svg20200612-01-l2" d="M 750.0 100.0 L 830.0 100.0"/>
  <path class="svg20200612-01-l2" d="M 10.0 210.0 L 290.0 210.0 L 290.0
    390.0 L 10.0 390.0 Z"/>
  <text class="svg20200612-01-t1" x="30.0" y="230.0">4</text>
  <path class="svg20200612-01-l1" d="M 80.0 255.0 L 80.0
    300.0 L 125.0 300.0"/>
  <path class="svg20200612-01-l1" d="M 161.25 255.0 L
    206.25 266.25
    L 195.0 311.25"/>
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  <text class="svg20200612-01-t1" x="330.0" y="230.0">5</text>
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  <path class="svg20200612-01-l1" d="M 380.0 243.75 L
    520.0 266.25"/>
  <path class="svg20200612-01-l1" stroke-dasharray="5,5" d="M 520.0 266.25 L
    555.0 271.875"/>
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    450.0 356.25"/>
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  <text class="svg20200612-01-t1" x="630.0" y="230.0">5p</text>
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    820.0 255.0"/>
  <circle class="svg20200612-01-c1" cx="750.0" cy="255.0" r="2"/>
</svg>
</div>

---

# Common notions

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  <text class="svg20200612-02-t1" x="30.0" y="30.0">1</text>
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  <text class="svg20200612-02-t1" x="330.0" y="30.0">2</text>
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  <text class="svg20200612-02-t1" x="450.0" y="100.0">a + b = a + b</text>
  <path class="svg20200612-02-l2" d="M 610.0 10.0 L 890.0 10.0 L 890.0
    190.0 L 610.0 190.0 Z"/>
  <text class="svg20200612-02-t1" x="630.0" y="30.0">3</text>
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  <text class="svg20200612-02-t1" x="30.0" y="230.0">4</text>
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  <circle class="svg20200612-02-l1" cx="490.0" cy="300.0" r="26.666666666666668"/>
</svg>
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---

# Book I, Proposition 47

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  <path class="svg20181107-04-l2" d="M 34.0 66.0 L 84.0 66.0 L 84.0 116.0 L 34.0 116.0 Z"/>
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  <path class="svg20181107-04-l3" d="M 52.0 42.0 L 34.0 116.0"/>
  <path class="svg20181107-04-l3" d="M 84.0 66.0 L 10.0 48.0"/>
  <text class="svg20181107-04-t1" x="51.5" y="35.0">A</text>
  <text class="svg20181107-04-t1" x="29.0" y="71.0">B</text>
  <text class="svg20181107-04-t1" x="89.0" y="71.0">C</text>
  <text class="svg20181107-04-t1" x="29.0" y="121.0">D</text>
  <text class="svg20181107-04-t1" x="89.0" y="121.0">E</text>
  <text class="svg20181107-04-t1" x="5.0" y="48.0">F</text>
  <text class="svg20181107-04-t1" x="28.0" y="19.0">G</text>
  <text class="svg20181107-04-t1" x="76.0" y="5.0">H</text>
  <text class="svg20181107-04-t1" x="113.0" y="34.0">K</text>
  <text class="svg20181107-04-t1" x="52.0" y="121.0">L</text>
</svg>
</div>

---
class: large-points

# Direct results

- Euclidian geometry
- geometric algebra
- subtractive GCD
- square root of two: infinity
- infinite prime numbers
- Platonic solids (4, 6, 8, 12, 20) - not all numbers are the same

---
class: large-points

# Scolia

- the need for scolia in classic style
- only prove the difficult case (e.g. B.I P.7 has additional cases that require
  2nd part of B.I P.5)
- systematic exploration of a problem: the converse propositions, but also
  more complex cases like the congruence of triangles
- analysis and synthesis

---
class: large-points

# Logic and reasoning

- the historical benefit/coin story
- demonstrative proofs rather than empirical methods
- axiomatic system: avoid circular reasoning
- reduction to absurd
- definition: what it is and that it is: B.I P.46
- definition in the negative: B.I D.1
- definition in the logical vs intuitive order
- demonstration of things that cannot be done e.g. B.I P.7
- the power of reasoning: atoms
- the dangers of pseudo-reasoning: life on other planets
- the aesthetic choice in mathematics

---
class: large-points

# Various

- Zeno
- Diophantus
- al-Khwarizmi - the Arabs

---

# Hyperbolic (Poincaré disk)

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    216.0 216.0 0 0 0
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</div>

---

# Elliptic (Sphere geodesic lines)

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# Cantor

- 1, 2, 3, 4, 5, ...
- 1, 4, 9, 16, 25, ... (Galileo's paradox)
- 0, 1, 2, 3, 4, ... (natural numbers)
- 0, -1, 1, -2, 2, ... (integers)
- 1/1, 1/2, 2/1, 1/3, 2/3, 3/1, ... (rationals)
- ..., sqrt(2), ... (solutions of algebraic equations)

---

# Peano

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# -isms

- Paradoxes
- Logicisms
- Intuitionism / constructivism
- Formalism
- Consistency
- Wir müssen wissen – wir werden wissen

---

# Gödel's Incompleteness

---

# Turing's machine

---

# Stepanov's Elements (EOP)

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# About EOP

- Bjarne Stroustrup: "some of the most beautiful code I've ever seen"
- relatively short book
- in classic style
- lacks scolia
- the structure of STL (algorithms with some data structures)
- concepts
- multithemed
- recurring themes: efficiency, mathematical foundation of algorithms, etc.
- systematic exploration of the fundamentals of (generic) programming for a
  machine with a CPU and memory

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# EOP Contents

- Chapter 1
- Chapter 2
- Chapter 3
- Chapter 4
- Chapter 5
- ...
- Chapter 12
- Appendix A and B

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# EOP Concepts syntax

```cpp
template<typename I>
    requires(Readable(I) && Iterator(I))
I find(I f, I l, const ValueType(I)& x)
{
    // Precondition: readable_bounded_range(f, l)
    while (f != l && source(f) != x) f = successor(f);
    return f;
}
```

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# GCD in rotate

![Image](../assets/2020-02-26-irregularity/11-rotate.png)

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# Regularity/Irregularity

- the importance of Euclid's common notions
- equality and (strict total) order
- the complexity of equality
- stronger: identity
- weaker: congruence
- even weaker: same area
- actually the importance of relations

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# C++ concepts

- long history
- C++11
- EOP
- Redesign for C++20, usage: [Good concepts](https://www.stroustrup.com/good_concepts.pdf)
- Irregularity [SLE2011](http://www.stroustrup.com/sle2011-concepts.pdf)

---

# Questions?