Maths Discussion Thoughts

What is the role of intuition in mathematics? How about imagination?

I believe that although mathematical knowledge must be proven with rigorous proof, initial ideas are sparked by intuition and imagination.  A mathematician might choose to use certain proofs in order to prove their conjecture based on their own intuition, and then use logical steps to see if it would work.

How important is logic(al reasoning) within Mathematical knowledge?

I think that logical reasoning is required within mathematical knowledge because in order for a conjecture to be believed and be ‘true’ it must be proved with rigour and clear steps to show how one came to that conclusion.

How connected is Maths to the real world?

Although maths might be more involved in topics such as the natural sciences, I believe that it can be used as a measure for other values. Despite that, maths might not always be effective. For example, numerical values have started becoming more popular to measure one’s attractiveness. Many might say that beauty is subjective, however we must admit that there are some people or animals that we would consider “beautiful,” or “cute,” and therefore certain values could be set to determine who is more attractive compared to someone else.

 

James Wright Poem

Lying in a Hammock at William Duffy’s Farm In Pine Island, Minnesota BY JAMES WRIGHT

I think that the last line of the poem extended our knowledge and interpretations, rather than bringing the poem together. The last line was surprising, although once I read it, I was able to notice other small details in the poem that supported the conclusion. For example, in the first line of the poem, “Over my head, I see the bronze butterfly,” I noticed that the pronoun used was “the,” instead of “a,” which suggests that that the poet was more used to the setting than we originally thought, and that his observations were not necessarily random and that the last line proved that he had in fact “wasted [his] life.”

Jesus Hopped the A-Train

I think that this play is a dark comedy drama, and I believe that this is because within each conversation between the several main characters, the topics of justice and faith are brought up often, yet the language is less sophisticated. However, I believe that we were laughing because of what the characters were saying, especially the inmates, but more importantly why they were reacting in such ways. I found that whilst reading this play I grew to like both Lucius and Angel despite the fact that they were both murderers. I think this is because the author initially introduces Lucius without mentioning why he is in custody, although later reveals why. The depiction of both characters is perhaps different to what I was expecting, considering their background, and is probably why I was more concentrated on their disagreements and petty threats rather than immediately disregarding them based on their crimes. Lastly, the more obvious reason would most likely be the vast use of vulgar language.

AMK Minds Progress Reflection 2

LO1 – Awareness

What previous experiences have I had with an activity like this?

What are my strengths and what would I like to improve?

Why have I struggled in some areas?

LO3 –  Initiative

What activities did we plan?

How did our plans as the activity progressed? Why?

What difficulties did we face in executing our plan? Why?

How did we overcome these challenges?

 

 

The Mathematical Method

Of the methods trying to prove the conjecture “the internal angles of a triangle sum to 180º ”, explain why only one provides ‘rigorous proof’.

Out of the three method uses to prove this conjecture, I think that two of them provided proof, whilst the other showed ‘rigorous proof’. The two that showed proof were the ones that took the edges of a triangle and either cut or folded them to create a semi circle, which represents 180º. These methods did not show rigorous proof because it did not demonstrate an unbroken chain of steps that led to this conjecture. Furthermore, there was a lot of human error involved within these methods which questioned the reliability of the proof. The third method that provided rigorous proof was the one that used three axioms to prove this conjecture. However, these axioms were not proved themselves but are just universally agreed truths that we came to believe by imagining different situations. Additionally, by adding these axioms together, there was a continuous chain of reasoning that proved this conjecture to be true, and something that we could not question.

Explain the difference between ‘proof’ and ‘rigorous proof’, using the SHIP -> DOCK example.

In the SHIP -> DOCK example, all the intermediate words contained at least one vowel, which could not be proven using experimental evidence. This is because no matter how many words we found that demonstrated this, there would always be the possibility of more words that could also. This example demonstrates rigorous proof as there are steps to show that all intermediate words do contain at least one vowel. Firstly, we have to accept and acknowledge that all “valid” English words contain a vowel. Secondly, the intermediate words between SHIP to DOCK must at some point have two vowels as only one letter at a time can change. In order for the previous statement to be false, the vowel in position three has to become a consonant and the consonant in position two has to become a vowel in one step.  However, this involves two letter changes which is impossible as only one letter can change at a time, thus all intermediate words must contain a vowel.

How does the term proof apply differently in maths and the natural sciences?

I think that proof in mathematics is much more rigour than in natural sciences, and this is because scientific knowledge is at first discovered and then tested, and referred to as “evidence.” Whilst, mathematical knowledge is something that can be seen and proven at any time and will always be correct, especially with the use of rigorous proofs where steps are clearly laid out. I believe that in natural sciences, we cannot always see things that we claim to be true, but we have different examples and evidence to back up scientific knowledge.

Where can maths be ‘found’ in nature?

  1. Honeycombs – Bees can easily create the hexagonal shapes found in honeycomb, whilst it would require a lot of effort for humans to recreate the shapes. Honeycomb demonstrates a repeated pattern that covers a plane, similar to mosaics or tiled floors. Mathematicians  have suggested that this shape is created in order to have to an efficient and large storage for honey with the use of minimal wax.  For example, circles would have spaces between each shape and therefore have less efficient storage of honey. However, some believe that the symmetrical shape of honeycombs is accidental and that bees would never been able to perform such an intricate task.
  2. Faces – Human faces have bilateral symmetry, that some believe is an aspect that determines physical attraction. Research has shown that mouths and noses are placed at “golden sections” of the space between the eyes and the chin. A spiral shape is formed by the comparable proportions from the side of the face. Statistics have shown that averages are close to the value of phi, and that it is believed that the closer the proportions are to phi, the more attractive one is perceived to be. Some say that it is possible that we as humans are designed to comply with the “golden ratio” as it promotes reproductive health.
  3. Starfish – Starfish have bilateral symmetry, however they can show radial symmetry through the process of metamorphosis, where the organism that be divided into halves. Starfish have at least five limbs, which can form ‘pentraradial symmetry’. However, this symmetry has been inherited and slightly modified through evolution from their previous ancestors.

Briefly explain why Galileo may have said: “Nature’s grand book, which stands continually open to our gaze, is written in mathematics.”

When I first saw read this statement I was initially unsure about what it meant and in fact I still am, but from what I understand about it I do agree. First of all, Galileo states that what we know about Nature, will always be “continually open to our gaze,” meaning that perhaps there is always more that can be discovered about nature and we can never truly know everything. The most significant and potentially controversial part is Galileo’s belief that “Nature’s grand book, is written in mathematics.” I personally believe that mathematics is more discovered than invented as although humans have invented units and numbers, they are just values that represent life. We use these values to explicitly show why or how something might work, but in order to explain different concepts mathematics is vital and had to be initially discovered in order for humans to explain what we know. I believe that this statement is true as mathematics is constantly demonstrated through the natural sciences. However, this concept is perhaps difficult to show as with Science there is always the question of whether there are examples that will falsify this pattern. Although there is still so much that we don’t know, I think that in nature, maths in the reason why we explain why different processes happen and why organisms grow to show different characteristics.

 

 

AMK Minds Progress Reflection

The local service that I am currently apart of is called AMK Minds. It is an organisation that works with mentally disabled people from ages 18 and above in order to not only support them but to also improve their long-term wellbeing.

I chose to be apart of this service as I have not worked with people with mental illnesses before and I felt that by taking part in this service, I would have the opportunity to challenge myself in regards to my interaction with others and how I might choose to go about doing that. I am aware of the challenges that many come in this service as I think that no matter who you work with, it is important to cater our actions based on their needs. However, although we may hope to make a long-term significant impact, small changes and improvements in their daily lives would be a much more sustainable and successful goal.

Throughout the course of my participation in this service, I hope to gain insightful knowledge about working with others and how I can critically think to generate activities that would be suit the client. Although our involvement in this service should be a rather selfless act, I do hope to gain some sense of fulfilment by the end, as knowing that I have made others feel slightly better even just for a day, does in fact make me feel somewhat contented.

The season has just begun and so we have actually not met with our service yet, as we are currently in the planning and preparation stage. However, we did meet with a couple of the organisation’s leaders in order to ask questions and learn about how we should approach the clients and any challenges that we could face. Our main aim is to get the clients engaged and active for the hour that they spend with us, and to ensure they are having fun and feel satisfied by the end of the session. In order to achieve this, we have planned several activities that we could run in order to keep them active but also to encourage interaction between both us, the service group, and them.

How we “know” and with what “certainty” differs across Areas of Knowledge

After a few weeks of TOK, I have come to believe that we can never be truly certain of the things we claim to know. Across the different areas of knowledge there are perhaps different ways of knowing that might seem more reliable than others. For example, in Natural Sciences we tend to base our knowledge of off scientific studies and evidence that has been gathered for many years. Although, it is important to think about the extent to which we should let our knowledge be influenced by scientists. For example, after my brief research on the Piltdown Hoax, I have learnt that although something is believed by many to be true, it does not mean that it is. However, at the same time I think that we cannot be completely certain of the things that we know, because in order to do so we would perhaps have to see it ourselves, which can be impossible to do in most cases.  An example would be Religion, where just because we cannot see something does not mean it isn’t true. Although in some areas, such as the Arts and Ethics, I believe that what we know about them is based more on interpretation and imagination, and that there are not “correct” answers but there are more socially accepted beliefs. Furthermore, just because knowledge from Natural Sciences tends to be based more evidence, and has been proven using the scientific method and processes of falsification and repetition, it does not mean that the quality of knowledge from it should be seen as of a higher quality than Ethics for example.

ALIS Test

To what extent can you rely on these results to accurately predict your final IBDP grades?

I think that only using these results to determine our predicted grades would be slightly inaccurate as it is just one test on one day, which might not have gone as well as you hoped for. I also think that the design of the test might not be the best way to measure the knowledge of someone as it tests only certain skills under pressure.

What process do you think we should use to come up with your final predictions?

I believe a good amount of factors should be considered when coming up with our final predictions. These would include our end of year exams, our class assessments, and what level the teacher honestly thinks we are at.

What are the strengths and weaknesses of knowledge acquired in this way?

I particularly did not enjoy this test as much, especially because of the time limit. This is because I felt that I had the ability to complete the questions successfully, but due to the time limit I was more panicked and so ended up answering some questions without thinking through it properly.

To what extent is intelligence fixed?

I don’t think that intelligence is fixed because I believe that we can always expand our knowledge and learn more. However, there are several different types of intelligences and I believe that some might be harder to control than others.

The Scientific Method

  1. Adaptation -> There is an interesting history of the concept of adaptation before Darwin’s research was brought about. In natural theology, adaption was “interpreted as the work of a deity and as evidence for the existence of God.” Several people had certain views of adaptation that “shadowed” others. Leibniz, a German mathematician, had believed that God had introduced “the best of all possible worlds.” However, Charles Darwin had falsified this and emphasised the “flaws and limitations” which existed in the world of different organisms.
  2. Animal Echolocation -> The term of echolocation was first invented by a zoologist named Donald Griffin, who worked to demonstrate this concept with bats in 1938. Griffin had pointed out that an Italian scientist, Spallanzani, had performed a series of experiments that showed that “when bats fly at night, they rely on some sense besides vision.” Later on, a Swiss physician, Jurine, repeated the experiments done by Spallanzani, and concluded that the other sense that bats used at night was hearing. The production of scientific knowledge in this case was done by the process of repetition as other scientists build upon past research done by others in order to prove this concept.

“Adaptation.” Wikipedia, Wikimedia Foundation, 28 Aug. 2019, en.wikipedia.org/wiki/Adaptation#History.

“Animal Echolocation.” Wikipedia, Wikimedia Foundation, 7 July 2019, en.wikipedia.org/wiki/Animal_echolocation#Early_research.

 

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