Paradojas en física y matemáticas

 ¿Las conoces? 

Zeno

Russell

Barber

Liar

Sorites

Monty Hall

Birthday

Banack Tarski

Twin

Grandfather

Causal & logical possibility excercises

1. It's logically possible that 2 + 2 = 5

2. Is is logically possible for Fidel Castro to become president of the United States in 1959?

3.  It's logically possible for a deductive argument to have true premises and a false conclusion. 

4. It's causally possible to travel back in time.

5. Is it causally possible for two parallel lines to meet at an infinite point?

6. Is having four sides in a square logically necessary?

7. It's not causally possible for an animal that lacks a vertebral column (or backbone) to be a vertebrate.

8. Here's a problem sample: 

It's probably true that whenever (A) a baby is born, (B) someone somewhere in the world will die on the same day. But this is hardly surprising given the number of people dying and being born each day. Any connection between the two is purely an accident.

In order to understand if the connection between A and B is an accident, it is essential to consider a counterexample. Can you think of that counterexample? 

9. Is it causally possible for a father not to be a parent? 

10. Is the presence of oxygen causally necessary for the proper functioning of the brain?

11. Being an adult of over 18 years old is legally sufficient for having the right to vote.

12. The presence of a witness is legally necessary for a valid marriage.

Homework 3 (causation, fallacies & consensus)

PREGUNTA SOBRE CONSENSO

1. De acuerdo con el químico alemán Justus von Liebig, la fermentación de la sangre produce gases tóxicos responsables de las más variadas enfermedades, como el cólera, viruela y sífilis; el aire transportaba estos gases que se combinaban con otros provenientes de la descomposición de la materia orgánica en la Tierra.

a) ¿Qué nombre recibe esta teoría? 

b) ¿En qué año es defendida por el consenso médico? 

c) En las ciencias médicas la teoría miasmática fue desechada a finales del siglo XIX en favor de la teoría microbiana de la enfermedad. ¿En tu opinión, qué indica este cambio de consenso médico entre finales del siglo XIX y el siglo XX? 

d) Imagina que eres un historiador de la ciencia médica tratando de explicar la diferencia entre el consenso de la teoría miasmática y la microbiana. Hazlo en un párrafo. 

PREGUNTAS SOBRE CAUSACIÓN

2. Read the following paragraph and answer the questions below:

A powerful tsunami destroyed the Fukushima Daiichi Nuclear Plant’s power supply and cooling systems, causing three reactors to melt and spew large amounts of radiation. Highly contaminated cooling water applied to the damaged reactors has leaked continuously into building basements and mixed with groundwater. Initial reports of casualties following the tsunami at Fukushima’s power plant put the death toll in the hundreds, with hundreds more missing. 

Was the tsunami the cause of the Daiichi Nuclear Plant’s accident? Yes, No? Explain your answer one way or the other. 

3. Read this short NPR article. 

We find this declaration: 

“We have no evidence that the crew did anything wrong,” Homendy said.

(a) Does that mean there’s no human cause linked to the accident? Explain. 

(b) Are safety regulations, in this case, solely dependent upon human error? Explain. 

Read this paragraph: 

The overheating triggered an alarm, which caused the train’s engineer to immediately apply the brakes to bring the train to the stop. An automatic emergency braking system also came into effect. 

Which of these two caused the train to stop, the overheating or/and the automatic emergency breaking?  

(a) The overheating caused the train’s engineer to immediately apply the brakes to bring the train to a stop, or (b) simultaneously, “an automatic emergency braking system also came into effect? 

Read this sentence: 

“There were no reported fatalities or injuries caused directly by the derailment, and responders were able to mitigate the fire within two days.” 

What do you think “caused directly by the derailment” means? 

Read this paragraph: 

“The evacuation order was lifted on Feb. 8, five days after the derailment. Since then, residents have reported widespread concern about the safety of East Palestine’s air and water. Some say they have experienced headaches or rashes. The Ohio Department of Natural Resources reported that some 3,500 fish had died following the derailment in nearby waterways.” 

Do you think the headaches and rashes are the effects of the accident? If so, what is the cause (as given in the reading? 

Read this paragraph: 

“The operators of a Norfolk Southern train carrying hazardous chemicals near East Palestine, Ohio, earlier this month tried to stop the train after a wheel bearing overheated to a dangerous degree, the National Transportation Safety Board said.” 

Can one say that “trying to stop the train after a wheel bearing overheated to a dangerous degree” is the cause of the derailment?

EJERCICIO SOBRE FALACIAS

1. El ataque de Thomas Elder a la asistencia social no debe tomarse en serio. Se dice que era un degenerado.

2. Quedarás embarazada si tienes relaciones sexuales. Sally tuvo relaciones sexuales una vez y ahora está embarazada.

3. Su Señoría, el acusado debe ser culpable porque nadie puede probar su inocencia.

4. Gusto peculiar: o se tiene gusto o no se tiene.

5. Todos los hombres son iguales: criaturas egoístas que pasan cada fin de semana pegados a la pantalla del televisor.

6. Estoy seguro de que estará de acuerdo en que todos debemos trabajar juntos para aumentar las ganancias de esta empresa. Cualquiera que no esté de acuerdo con esta política será despedido de inmediato.

7. Todos creen que la Universidad de Lander mejora sus programas académicos cada año; por lo tanto, concluyo que Lander será una universidad mucho mejor valorada dentro de unos años.

8. Theodore Teabody, el gran historiador, dice que el libro de H. Beard es el mejor libro de lógica del mundo. Le creo, pues la palabra de un hombre tan grande no debe ser cuestionada.

Consensus proven wrong (proves paradigm shifts in science)

1543 – The transition in cosmology from a Ptolemaic cosmology to a Copernican one.

Two observations seemed to support the idea that Earth was the center of the Universe: First, from anywhere on Earth, the Sun appears to revolve around Earth once per day. While the Moon and the planets have their own motions, they also appear to revolve around Earth about once per day. Second, Earth appears to be stationary from the perspective of an earthbound observer; it feels solid, stable, and still. That's empirical evidence, isn't it?

Why It Worked For Ancient Science?

Daily Motion: The Sun, Moon, and stars visibly cross the sky in a predictable pattern.
No Sensation of Motion: Humans do not feel the Earth spinning or moving through space.
The Conclusion: Without instruments like telescopes, the data pointed directly to a stationary Earth at the center of the cosmos.

Why It Changed Later?

New Tools: The invention of the telescope provided new empirical evidence.
Better Observations: Galileo saw moons orbiting Jupiter, proving not everything revolves around Earth.
Parallax: Scientists eventually measured tiny shifts in star positions caused by Earth moving around the Sun.

1543 – The acceptance of the work of Andreas Vesalius, whose work De humani corporis fabrica corrected the numerous anatomical errors in the previously-held system created by Galen.

Why did Galen's views work for centuries?

• Apparent Accuracy: The ancient Greek physician Galen was a brilliant researcher whose comprehensive textbooks seemed flawlessly logical.
• Hidden Flaws: Roman law banned human dissection in Galen's era, so he extrapolated human anatomy from the dissection of animals such as monkeys, apes, and pigs.
• The Conclusion: Because no one was double-checking his work on actual human bodies, the medical world accepted animal anatomy (like a multi-compartment uterus or a multi-boned lower jaw) as human truth for over a millennium.

Why It Changed Later?

• Hands-On Dissection: Vesalius insisted on picking up the scalpel himself rather than reading to students while a servant did the cutting.
• Exposing the Truth: Direct observation allowed Vesalius to prove Galen had been "fooled by his monkeys". He corrected hundreds of fundamental errors, proving that the adult human jawbone is a single unit and humans do not have an extra pair of ribs.
• Visual Proof: Vesalius utilized master artists to create highly detailed, accurate woodcut illustrations. This visual evidence made his empirical findings impossible to ignore and effectively launched modern medicine.

1687 – The transition in mechanics from Aristotelian mechanics to classical mechanics.

According to Aristotle, there are four elements that make up everything in the terrestrial spheres: earth, air, fire, and water (Aristotle borrows this from Empedocles, proving there's already a consensus on this matter. Aristotle held that the heavens are made of a special weightless and incorruptible, unchangeable fifth element called "aether". For your information, "aether."

Why the Previous Aristotelian System Worked For Centuries

• Apparent Accuracy: Aristotle argued that an object requires a continuous force to stay in motion. If you stop pushing a cart, it stops moving. This matches everyday human experience.
• Hidden Flaws: Aristotle did not recognize friction and air resistance as external forces. He assumed objects naturally resisted movement and wanted to return to a state of rest.
• The Conclusion: Because friction is invisible and omnipresent on Earth, Aristotle's idea that "motion requires a constant pusher" seemed like undeniable empirical truth for nearly 2,000 years.

Why It Changed Later?

New Tools: The invention of the telescope provided new empirical evidence.
Better Observations: Galileo saw moons orbiting Jupiter, proving not everything revolves around Earth.
Parallax: Scientists eventually measured tiny shifts in the positions of stars caused by Earth moving around the Sun.

1718-present. – The transition from the luminiferous "aether" present in space to electromagnetic radiation in spacetime.

Go to this Wikipedia article.  I want to contrast the following statements by leading scientists:

Newton: "Doth not this aethereal medium in passing out of water, glass, crystal, and other compact and dense bodies in empty spaces, grow denser and denser by degrees, and by that means refract the rays of light not in a point, but by bending them gradually in curve lines? ...Is not this medium much rarer within the dense bodies of the Sun, stars, planets and comets, than in the empty celestial space between them?"

Einstein: We may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time. 

One more: Most physicists in the 20th century concluded that this early modern notion of a luminiferous aether was not a useful concept (Einstein however stated that this consideration was too radical and too anticipatory and that his theory of relativity still needed the presence of a medium with certain properties). There is simultaneously consensus and disensus. So is there an "aether" after all? We could call this an unsettled case in scientific endeavor.

1780s – The acceptance of Lavoisier's theory of chemical reactions and combustion in place of phlogiston theory, known as the chemical revolution.

Why the Phlogiston System "Worked"

The phlogiston theory worked for nearly two centuries because it was highly practical, deeply intuitive, and backed by everyday empirical evidence. It gave 18th-century scientists a single, unifying framework to explain three completely different phenomena: burning, breathing, and rusting. To an observer at the time, the system made perfect sense based on direct observation:

• It matched raw visual evidence: When wood burns, it visibly shrinks into a tiny pile of ash. The most logical conclusion was that a substance (phlogiston) was leaving the wood and escaping into the air.
• It successfully predicted reactions: The theory correctly predicted that a candle would go out in a sealed jar. The explanation—that the air became "choked" or saturated with phlogiston and couldn't hold any more—perfectly matched the physical reality of the flame dying.
• It explained respiration: why animals died in enclosed spaces (they filled the air with toxic phlogiston from breathing) and why plants refreshed the air (they absorbed the phlogiston).

Why It Changed Later?

The phlogiston system changed because new, ultra-precise measurement technology exposed a fatal flaw that the old theory couldn't logically explain. 

When chemists began using highly accurate scales to weigh substances before and after combustion, they ran into a massive contradiction: when metals were burned, the resulting ash weighed more than the original metal.

If burning meant losing phlogiston, the leftover material should have been lighter. To keep the old system alive, defenders had to invent increasingly absurd workarounds—even claiming that phlogiston had "negative weight" and floated upward. Antoine Lavoisier forced the change by completely flipping the perspective.