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poll questions

definitions change … shyness vs social anxiety disorder

multiple definitions for the same thing … RED is

colour of fire, colour of blood

electromagnetic radiation, in visible light spectrum, wavelength of 620-750 nanometres

hexadecimal: #FF0000

RGB (red, green, blue); (255,0,0)

[in a computer, colours are represented by numbers, and are manipulated using numeric algorithms (e.g., effects in photoshop)]

Photoshop, Illustrator

Web page design

Even “obvious” needs to be defined – what is milk?

optional reading: https://www.theguardian.com/lifeandstyle/shortcuts/2017/jul/23/dairy-milk-court-animal-plant-nut

Real world definitions depend on:

Time (ADHD, SAD, Pluto is no longer a planet)

Authority (freedom fighter, terrorist)

History (young, adult, teenager)

Geography (driving under influence)

Culture (acceptable behaviour, nudity)

Etc.

We must know things … to be aware, to make good decisions

Another example:

Initially, when cell phones came out, their makers strongly denied any effects of cell phone radiation to human health

However …

optional reading: http://www.dailymail.co.uk/health/article-3227648/Mother-kept-phone-bra-day-10-YEARS-convinced-caused-terminal-breast-cancer.html

iPhone:

SAR limits for iPhone 5

Previous page source:

https://www.sportportactive.com/knowledge-base/iphone-safety-guidelines-rf-exposure-information/

Read more about RF exposure, if you wish:

https://www.cancer.org/cancer/cancer-causes/radiation-exposure/cellular-phones.html

Problem

Hot cocoa: 99.9% caffeine free, in an 8 oz serving

Problem: Create a mathematically sound argument showing that Starbucks Venti (20 oz) blonde roast cup of coffee (which has 475 mg of caffeine) can be claimed to be 99.9% caffeine free.

[Source of caffeine information: Starbucks Nutrition Data https://www.starbucks.ca/menu/nutrition-info]

A (mathematical) theorem is a statement that establishes a new relationship between previously defined mathematical objects and/or their previously established properties.

Logical structure of a theorem is called implication: IF assumption(s) A THEN conclusion(s) B

In real life “If A then B” could be:

“If cause A then effect B” “A causes B” “A is B” or “A are B” “B is caused by A” “B is a consequence of A” “B follows from A”

Theorem establishes causation:

assumption(s) = cause conclusion(s) = effect

structure: cause implies effect

examples:

“a cat is a mammal” “meningitis causes headache” “destruction follows an earthquake” “sunburn is caused by UV exposure”

“smoking causes lung cancer” “disease implies symptom(s)”

in real life, we accept implication/causation even if it is not confirmed in absolutely all cases (so it is a causation with some – usually high – probability)

How do we use a theorem? •  if assumption(s) hold, then conclusion(s) is/are valid •  if some assumption does not hold, the conclusion(s) might hold or not hold

Converse of A implies B is B implies A

Equivalence: both A implies B and B implies A hold

Negative of A is not A

Contrapositive of A implies B is not B implies not A

To be accepted (and added to the theory) a theorem requires mathematically acceptable evidence (called proof)

Once a proof is established and accepted (i.e., correct), theory continues building

Math theory cannot be built on conjectures, hypotheses, or partially correct statements

What is acceptable evidence in science other than math?

Major evidence is obtained by replicability/ reproducability (same or similar results obtained by repeated experiments)

many issues: •  how many repeated experiments? •  what if some do not give same/similar results? (we use statistics; generally accepted in many situations: “19 out of 20 cases”)

Source (optional): https://www.ctvnews.ca/sci-tech/totally-bummed-nobel-prize-winner-admits-a-painful-mistake-1.4751683

Randomized control trial (RCT): pick a random sample, divide into treatment and control groups, and compare outcomes

Major question: how big a difference between the outcomes for the treatment and the control groups has to be to demonstrate that the treatment works?

RCT is one essential part of evidence-based medicine (which is a relatively new phenomenon! 1960s and on)

Evidence stronger than RCT:

•  meta-analysis (combines the results of multiple studies)

•  systematic review (complete, exhaustive summary of current evidence, based on literature search and secondary data)

For example, a systematic review contributed to establishing the fact that smoking causes lung cancer

Replicability/ reproducibility is not always possible:

Cosmology (the study of our universe) is based on one experiment – our universe! We are not aware of any other universes

Therefore … evidence:

* In math: proof (nothing else accepted)

* In Science, medicine, etc. established methods: replicability/reporoducibility, randomized control trial, meta-analysis, systematic review

* Elsewhere: chaos; what is acceptable evidence varies based on politics, culture, beliefs, etc.

“Evidence” based on coincidence:

antibiotics treat ear infections in children

New ear infection rule in many cases: give no antibiotics, just wait! (Note: we did not say … no antibiotics at all)

optional http://www.parents.com/health/ear-infection/the-new-ear-infection-rules/

So, to provide good evidence we must:

show that intervention (antibiotic) indeed has an effect (ear infection gone)

AND

show that when there is no intervention (no antibiotic given), there is no effect (ear infection persists)

this is hard, so …

So, to provide good evidence we must:

show that intervention (antibiotic) indeed has an effect (ear infection gone)

AND

show that when there is no intervention (no antibiotic given), there is no effect (ear infection persists) IN MOST CASES

standard: no effect in 19 out of 20 cases (we will connect this with the so-called p-value)

Questions we must ask: Is there evidence? What is acceptable as evidence? What exactly does the evidence show?

optional reading: http://www.cbc.ca/news/health/8-glasses-of-water-a-day-an-urban-myth-1.1196386

[http://www.nzherald.co.nz/lifestyle/news/article.cfm?c_id=6&objectid=11749101]

optional reading: http://news.bbc.co.uk/2/hi/health/8132857.stm

‘Because it sounds good’ is not evidence

“Evidence” based on authority, celebrity, popular culture – promoting pseudoscience

http://www.macleans.ca/society/dont-blame-gwyneth-paltrow/

Source (12 Jan 2020, optional): https://bit.ly/2tSzi54

… and in many cases – no evidence at all

Major example:

•  fake news •  deepfake (videos)

Find Math 2UU3 web page Click on the link for the online clickers, login, and reply to the

poll questions

We have definitions, theorems (implications), what else?

Conjectures, hypotheses, “working assumptions”

Example from math: Goldbach conjecture

Historic events (especially those preceding written culture) are based on someone’s recollection (which could be the only source); or there could be a total absence of (reliable) information

If a fact in math depends on previously established facts, where does it all start?

Axioms

axioms we must accept, as there is no way to prove them – so, we have to believe!

all sciences are based on beliefs

Other math statements/objects: Algorithms

examples:

solving an equation (“move terms from one side to another”)

finding extreme values (or peaks, tipping points) of a function (or some phenomenon)

also …

Algorithms are mathematical routines that are used, for instance, to extract patterns from large data sets

optional reading: http://www.forbes.com/sites/kashmirhill/2012/02/16/how-target-figured-out-a-teen-girl-was-pregnant-before-her-father-did/#6091340b34c6

Pole = Andrew Pole, Target Statistician

Father’s reply:

The words algebra and algorithm come from the same person

Abu Abd Allah Muhammad Ibn Musa al-Khwarizmi (ca 780- 850 CE)

Persian mathematician, astronomer, astrologer, geographer and author, worked in Baghdad

Al-Khwarizmi wrote Book of Completion and Balancing

defines algebra (al-jabr … subtracting a quantity from one side of the equation and adding it to another)

the word algorithm is derived from the latinization of Al-Khwarizmi name, Algoritmi

Hidden algorithms … algorithms embedded in numbers

Your symbol is …

what’s special about credit card numbers? 4128 0033 2019 8008

look at digits in odd positions: 4128 0033 2019 8008 … multiply them by 2 8 4 0 6 4 2 16 0 … and add digits … 31

add digits in even positions … 1+8+0+3+0+9+0+8=29

add up the two … 31+29 = 60

the sum is always divisible by 10 !!!

change any one digit … 4128 0033 2015 8008




interchange two neighbouring digits 4128 0032 3019 8008




… called Luhn algorithm

upc code

isbn

Luhn algorithm is an example of an error detecting code

credit card number

...

QR code

challenge …

create a valid visa credit card number!!!

4500 0301 _ _ _ _ _ _ _ _

(Aeroplan Infinite Visa Card)

reality …

optional reading: http://www.cbc.ca/news/technology/algorithms-hiring-bias-ramona-pringle-1.4241031

Serious problems with algorithms and AI (= artificial intelligence): machines learn from humans!

More examples of algorithms

* all of artificial intelligence

* encryption/decryption of information sent over internet

* driving directions; autopilot

* social networks (news, advertising, linking)

* search engines

* trading (financial markets)

Formulas use data to obtain results, hopefully meaningful

Wind chill – based on air temperature and wind speed (describes feeling of being cold)

Humidex – based on air temperature and humidity

BMI = body mass index

UV index = intensity of our Sun’s radiation which reaches us

How to think about a fraction:

AB

if A increases (B fixed), then the fraction increases

AB

if A decreases (B fixed), then the fraction decreases

AB

if B increases (A fixed), then the fraction decreases

AB

if B decreases (A fixed), then the fraction increases

but …

Example: body mass index

BMI = mass [kg]height [m]( )2

Of two people with the same mass, the taller one has smaller BMI

Of two people of the same height, the heavier one has a larger BMI

When numbers make no sense

9-year old girl Ana

Optional video: http://www.cnn.com/video/data/2.0/video/us/2014/05/26/mxp-wabc-bmi-school-program-girl-overweight.hln.html

According to her school administrators, Ana is overweight …

height (m)

weight (kg)

BMI

1.24 28 1.25 30 1.27 31 1.30 31 1.33 32

NORMAL GROWTH PATTERN FOR KIDS

height (m)

weight (kg)

BMI

1.24 28 1.25 30 1.27 31 1.30 31 1.33 32

Ana: height = 4 ft 1 in = 1.25 m weight = 66 lb = 29.94 kg = 30kg

height (m)

weight (kg)

BMI

1.24 28 1.25 30 19.2 1.27 31 1.30 31 1.33 32

BMI = weight in kg divided by height in m squared

i.e., BMI = mass [kg]height [m]( )2

height (m)

weight (kg)

BMI

1.24 28 18.2 1.25 30 19.2 1.27 31 19.2 1.30 31 18.3 1.33 32 18.1

BMI = weight in kg divided by height in m squared

i.e., BMI = mass [kg]height [m]( )2

height (m)

weight (kg)

BMI

1.24 28 18.2 normal 1.25 30 19.2 overweight 1.27 31 19.2 overweight 1.30 31 18.3 normal 1.33 32 18.1 normal

It is well known that patterns of child growth vary significantly – there is no simple relation (such as the one expressed in BMI) between height and weight

Reasoning based on BMI does not apply to children

Evidence: [optional reading: Cogan, Smith, Maine (2008). The risks of a quick fix: a case against mandatory body mass index reporting laws. Eat Disord. 2008 Jan-Feb;16(1):2-13. doi: 10.1080/10640260701771664.]

This article describes the state level activity on mandatory BMI reporting, offers numerous arguments against this practice, and suggests an alternative approach to promoting health in youth. Mandatory BMI reporting laws place a new and inappropriate responsibility on the schools. Proponents of such laws imply that BMI reporting will have positive outcomes, yet there is virtually no independent research to support this assumption. The authors argue that these laws could do significant harm, including an increased risk for children to develop eating disorder symptoms.

According to her school administrators, Ana is overweight … total nonsense, stupid, dangerous!

However,

New York City Education Department defended their fitness program, saying that it is “effective and helps students get healthy”

optional reading: https://ca.style.yahoo.com/many-calories-burn-while-walking-231202871.html

on the next slide we will write it as a math formula

Calories per minute = 0.035 ⋅B + 0.029 ⋅B⋅V2

H

B= body mass in kilograms V = velocity in metres per second H = height in metres

Calories per minute = 0.035 ⋅B + 0.029 ⋅B⋅V2

HB= body mass in kilograms V = velocity in metres per second H = height in metres

the taller you are, less energy it takes … how is it reflected in the formula?

Good to remember

Kcal means kilocalories; so 1 Kcal = 1000 calories

BUT NOT when food is concerned: in that case, 1 Kcal = 1 (food) calorie

When you travel, keep in mind: EU food labels sometimes use Kcal

Quantifiers and rules of formal logic

Universal: “for every”

Existential: “there exists”

Conjunction: A and B

Disjunction: A or B (inclusive or)

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