-
Notifications
You must be signed in to change notification settings - Fork 0
FrequentlyAskedQuestions
Isn't the triangle inequality pretty much the Pythogorean identity?
Not exactly, although they are both important concepts in mathematics.
-
The Pythagorean theorem is a statement about right triangles in Euclidean geometry: in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as a^2 + b^2 = c^2, where c is the hypotenuse, and a and b are the lengths of the other two sides.
-
The triangle inequality, on the other hand, is a statement about the lengths of any sides of any triangle, not just right triangles. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. In other words, if you have a triangle with sides of lengths a, b, and c, then: a + b > c, b + c > a, and a + c > b.
Both are foundational principles in geometry and other areas of mathematics, but they are different principles with different applications. For instance, the Pythagorean theorem is specifically about right triangles and is central to concepts like distance in Euclidean space. The triangle inequality is more general, and is used in a wide range of mathematical fields, including analysis, linear algebra, and topology, where it helps to define the concept of a metric or distance function.
Neither the Pythagorean theorem nor the triangle inequality directly address the reason a finite speed of light exists. The finite speed of light is a postulate of Einstein's theory of special relativity, which states that the speed of light in a vacuum is constant and is the same for all observers, no matter the motion of the light source.
The principle underlying this postulate is empirical in nature; it comes from a long history of experiments in physics. The Michelson-Morley experiment, for example, was crucial in establishing that the speed of light is constant in all inertial frames of reference, which later became one of the key postulates of special relativity.
The Pythagorean theorem and triangle inequality are mathematical principles that may be used in the mathematical formulation of physical theories, including those that describe light's behavior, but they do not, by themselves, explain why the speed of light is finite.
The terms used not only influence how one thinks and communicates, but also shape the understanding and perception of those learning from the research. "Perturbation", while accurate, may imply unwanted interference. By using 'stimulation', the intentional and controlled nature of the work is emphasized.
'Stimulation' signifies a precise input to study system response and is only 1 edit away from being 'simulation', which is commonly done to predict and forecast states of processes either consciously or unconsciously. This term more effectively describes the meticulous arrangement of ideas whilst simultaneously fostering a more positive and approachable understanding of the concepts involved.
In physics, the term "susceptibility" traditionally refers to a measure of how a system responds to an applied force or field. However, in this context, 'propensity' has been chosen, a term that carries a more proactive and positive connotation.
'Susceptibility' might unintentionally suggest passivity or vulnerability, whereas 'propensity' indicates an inherent characteristic of the system to respond, conveying a sense of agency and dynamism. This encourages a view of the system as an active participant in the experiments, not merely a passive recipient of external forces.
Moreover, introducing new terminologies can refresh the way established concepts are engaged with and prevents stagnation in thought processes. By using the term 'propensity', the aim is to inspire fresh perspectives and reinvigorate intellectual exploration in associated fields. Language can be as fluid and evolving as the phenomena under study, helping to foster continuous learning and discovery.
"CP variability" is suggested as a replacement for "CP violation" in particle physics. The rationale behind this is to more accurately depict the behaviors of elementary particles under charge conjugation and parity symmetry (CP).
"CP violation" described instances where these combined symmetries were unexpectedly not conserved. However, this implies an abnormal deviation from a rule.
A more nuanced understanding of particle behavior shows this isn't a "violation" in the negative sense, but rather an inherent property of these particles. The term "variability" better encapsulates this concept - particles display variable behaviors under CP transformations, not erratic "violations".
"CP variability" therefore acknowledges the complexity and variability in particle behaviors, and frames these as natural occurrences rather than exceptions. This shift in terminology underlines the scientific laws as models based on current understanding and observational capabilities, subject to evolution and refinement as more is uncovered about the universe.
When discussing advanced concepts in the realm of abstract algebra, terms such as 'structurally constant form' or 'structurally constant bilinear form' are sometimes preferred over the traditional term 'Killing form'. This choice reflects no disregard or disrespect for the German mathematician Wilhelm Killing, after whom the form is named. Rather, it's a conscious decision aimed at emphasizing the mathematical essence and construction of the form over its historical attribution.
The terms 'structurally constant form' or 'structurally constant bilinear form' immediately convey a significant aspect of the form – that it is intrinsically defined in terms of the structure constants of the Lie algebra. This name, therefore, encapsulates some of the mathematical significance and meaning of the form, which can be a powerful aid in understanding and communicating about the concept.
It's important to note that historical eponyms like 'Killing form' carry significant weight and are universally recognized within the mathematical community. However, the use of more descriptive names can sometimes enrich conceptual understanding and discussions, especially for those who are newly encountering these ideas. As always, when using alternative names, striving to maintain clear communication and respect for the traditional terminology that forms part of mathematical heritage is crucial.
First lets look at the definitions of the roots of the two words:
- synergize: to combine or coordinate the activity of (two or more agents) to produce a joint effect greater than the sum of their separate effects or the working together of two things to produce an effect greater than the sum of their individual effects
- entangle: entrap or twist together or entwine into a confusing mass;
The terminology used in quantum mechanics shapes our perceptions of the underlying phenomena. The term 'entanglement' has long been employed to characterize the interrelations and non-local nature of quantum particles. Yet, this term, originally associated with intricate traps and complex entrapments, might not best represent the phenomena it's meant to describe. Its negative connotations and the intricate web of confusion it spins do little to foster a comprehensive understanding of the complex and nuanced quantum correlations.
As our grasp of quantum mechanics deepens, the terminology should also adapt. The proposed replacement is 'synergy', a term rooted in system theory, implying that the effect of a whole system exceeds the sum of its parts. This offers a broader and more dynamic interpretation of quantum phenomena, removing the negative undertones associated with 'entanglement'. Under this new terminology, traditional phrases like ‘‘let A and B be entangled particles…'' would become ‘‘let A and B be synergized particles…" or 'let A and B be a pair of synergistic particles which were synergized via this method.... This shift in language more accurately depicts the complex, non-competitive, and harmonized interactions that are inherent to quantum mechanics.
Introducing 'synergy' to the quantum lexicon aims not to simplify, but to enrich and improve the expression and understanding of quantum phenomena. It encourages thinking about quantum mechanics in a way that is removed from the snare-like complications suggested by 'entanglement', promoting progress in the field and fostering a more constructive conceptual framework for future generations.