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?
- 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.
- 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.
- 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.