mathematics12

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Reaction on the movie A Beautiful Mind (2001)

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Who is John Nash?

John F. Nash, Jr., was an American mathematician who won the 1994 Nobel Prize in Economics, along with John Harsanyi and Reinhard Selten, for his development of the mathematical foundations of game theory. Nash was also a pioneer in the study of differential geometry and partial differential equations. He also developed an equilibrium theory known as the Nash Equilibrium.

Source: https://www.investopedia.com/terms/j/john-f-nash-jr.asp

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By the early 1970s, Nash received treatment that allowed his condition to improve to the point that he was able to begin teaching again at Princeton. It was there that he served as senior research mathematician for the last 20 years of his life. There, he later became known as the "Phantom of Fine Hall" for his habit of filling blackboards with complex equations during the night when no one else was around.

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Notable Accomplishments and Deeds

Among Nash's other groundbreaking math theories: the Nash-Moser inverse function theorem, the Nash–De Giorgi theorem, the Nash embedding theorems, which the Norwegian Academy of Science and Letters said were "among the most original results in geometric analysis of the twentieth century”.

Source: https://www.investopedia.com/terms/j/john-f-nash-jr.asp

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What is game theory? How does it impact our understanding of things in real life?

Game theory is used extensively in various forms of collective bargaining and negotiation. For instance, during a strike or lockout, unions and management negotiate to raise wages. It is possible to maximize the welfare of both workers and control by using game theory to arrive at the optimal solution.

Game theory is a framework for understanding choice in situations among competing players. Game theory can help players reach optimal decision-making when confronted by independent and competing actors in a strategic setting.

Source: https://upjourney.com/game-theory-examples-in-real-life

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Game theory was developed in the mid-20th century by economist Oskar Morgenstern and mathematician John von Neumann. The mathematical model seeks to determine the best option for an individual player when the other participant's unknown choice affects the overall outcome. This theoretical framework is more of a scientific strategy that yields optimal decision-making.

In the real world, game theory is applied when analyzing scenarios such as pricing competition and product development among competing firms. The game theory definition describes the concept as a model for understanding interactive situations among competing players. As such, the theory assumes that the strategy implemented by one player shapes the other player's payoff. Indeed, its proponents argue that the outcome of social interactions is influenced by different identities, strategies, and preferences of competing players. Nonetheless, different game theory models impose additional assumptions and requirements.

Source:https://study.com/learn/lesson/game-theory-overview-examples-application.html

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Game theory can determine likely results in situations involving multiple players with known outcomes and quantifiable payouts. This payout can be in the form of utility or financial payoff. Game theory provides a mathematical approach for analyzing situations that involve parties who make interdependent decisions. As already stated, the outcome of social interactions (game) depends on the optimal decisions of the interacting parties (players). It is important to note that the players may have identical, contradictory, or diverse interests.

Although game theory is a branch of applied mathematics, it has many applications. The concept can be applied in evolutionary biology, business, psychology, politics, and economics. Regardless of its unmistakable advances, the theory remains young, and many experts agree that it is a developing science.

Source: https://study.com/learn/lesson/game-theory-overview-examples-application.html

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Mental health awareness is the ongoing effort to reduce the stigma around mental illness and mental health conditions by sharing our personal experiences. Because of misconceptions and stigma surrounding mental health issues, people often suffer in silence and don't seek treatment for their conditions. Mental health awareness is an important initiative to improve understanding of mental health conditions and increase access to healthcare for those who need it.

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John Nash had a publicized fight with mental illness. Write down your thoughts on mental health. How do you take care of yourself during the ongoing COVID pandemic?

Mental health includes our emotional, psychological, and social well-being. It affects how we think, feel, and act. It also helps determine how we handle stress, relate to others, and make healthy choices.  Taking care of our emotional well-being can help us be more productive and effective at work and in our daily activities.

Healthy Ways to Cope with Stress during pandemic:

Take breaks from news stories, including those on social media. It’s good to be informed, but constant information about the pandemic can be upsetting. Consider limiting news to just a couple times a day and disconnecting from phone, tv, and computer screens for a while.

Take care of your body.  Eat plenty of fruits and vegetables, lean protein, whole grains, and fat–free or low–fat milk and milk products. Going to bed at the same time each night and getting up at the same time each morning, including on the weekends, can help you sleep better

Make time to unwind. Try to do some other activities you enjoy.Connect with others. 

Talk with people you trust about your concerns and how you are feeling.

Connect with your community- or faith-based organizations. While social distancing measures are in place, try connecting online, through social media.

Source: https://www.cdc.gov/mentalhealth/stress-coping/cope-with-stress/index.html

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Thanks for visiting!

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Incompleteness Theorem

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Kurt Gödel (1906-1978), who by 1931, at the age of 25, had published his two "Incompleteness Theorems." He became the enfant terrible in the world of formal logic.

To understand Gödel's impact in the logical world one must bear in mind the great project that for decades had been the program of modern logicians: to prove that all mathematics is founded upon logic and can be derived from a few basic logical axioms.

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What is Godel's Theorem (Henriksen, 1999)

Godel's Incompleteness Theorem asserts that there are properly posed questions involving only the arithmetic of integers that mathematics cannot answer nor precisely evaluate. Such assertions are called undecidable, and according to Godel, such assertions will always exist.

Even if you attempt to refine the system in which an undecidable statement is decreed true,another undecidable statement would be generated to take its place.

Source:https://www.scientificamerican.com/article/what-is-godels-theorem/

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Math's Existential Crisis (Undefined Behavior, 2016)

Mathematics is known for its logical precision, we can tackle any problem with absolute certainty, we don't have to deal with inaccurate measurements or subjective biases, but, math isn't as perfect as we hope.

Math system has the base of axioms that altogether build theorems with proofs that set its structure. Properties: Sufficiently expressive and completeness (ability to answer all known answers,system that proves everything within it), Consistency (can prove anything) and Math can be inconsistent.

First Incompleteness theorem: According to Godel, any sufficiently expressive math system must be either incomplete or inconsistent. We can't just add any unsubstantiatable axiom and call it a day, it doesn't work like that. Adding an axiom creates a new system that remains susceptible to the incompleteness theorem. There will be statements that still cannot be proved. We are forced to accept that the system is incomplete. Godel's second incompleteness theorem: A consistent math system cannot prove its own consistency. We prefer our systems to be incomplete rather than inconsistent, being unable to prove its consistency is an example of its incompleteness. Math isn't perfect and it won't give us all answers, they may never be an end. We may have been using faulty methods while coming up with the system. We may feel disappointed when learning these theorems but it leaves room for exploration.

Source: https://www.youtube.com/watch?v=YrKLy4VN-7k

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Mathematics (Knorr et al., 2020)

Mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter.

All mathematical systems (for example, Euclidean geometry) are combinations of sets of axioms and of theorems that can be logically deduced from the axioms. Inquiries into the logical and philosophical basis of mathematics reduce to questions of whether the axioms of a given system ensure its completeness and its consistency.

Source: https://www.britannica.com/science/mathematics/Ancient-mathematical-sources

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Incompleteness Theorem (Hosch, 2011)

This theorem established that it is impossible to use the axiomatic method to construct a formal system for any branch of mathematics containing arithmetic that will entail all of its truths. In other words, no finite set of axioms can be devised that will produce all possible true mathematical statements, so no mechanical (or computer-like) approach will ever be able to exhaust the depths of mathematics.

It is important to realize that if some particular statement is undecidable within a given formal system, it may be incorporated in another formal system as an axiom or be derived from the addition of other axioms. For example, German mathematician Georg Cantor's continuum hypothesis is undecidable in the standard axioms, or postulates, of set theory but could be added as an axiom.

The second incompleteness theorem shows that a formal system containing arithmetic cannot prove its own consistency. In other words, there is no way to show that any useful formal system is free of false statements. The loss of certainty following the dissemination of Gödel's incompleteness theorems continues to have a profound effect on the philosophy of mathematics.

Source: https://www.britannica.com/topic/incompleteness-theorem#ref1107613

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Gödel's Incompleteness Theorems (Ditts, 2017)

Mathematics tries to prove that statements are true or false based on these axioms and definitions, but sometimes the axioms prove insufficient. Sometimes the axioms lead to paradoxes, like Russell's paradox, and so a new set of axioms are needed. Sometimes the axioms simply aren't enough, and so a new axiom might be needed to prove a desired result.

Gödel's incompleteness theorems show that pretty much any logical system either has contradictions, or statements that cannot be proven!

Source: https://infinityplusonemath.wordpress.com/2017/08/04/godels-incompleteness-theorems/