The world of physics and engineering is about to make a quantum leap as IBM prepares to launch the first commercially available quantum computer in the cloud. This article provides a detailed explanation of what quantum computing is all about.
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
UNDERSTANDING MATHEMATICAL SYSTEMS
Before we question the incompleteness or inconsistency of mathematics, let us understand the structure of a mathematical system. There exists several system such as decimal arithmetic and binary arithmetic, just to name a few. Each of these systems is built upon the most fundamental assumptions/truths known as axiom. For e.g. in binary arithmetic, we know A.A = A. Each mathematical system has a boundary of its own – meaning, we cannot use axioms outside the system boundary and expect things to work out.For e.g. the axiom A.A = A holds only for binary system and cannot be used during proving a theorem in the decimal domain.
UNDERSTANDING HOW MATHEMATICAL SYSTEMS ARE BUILT ( THROUGH AN ANALOGY)
At its most fundamental level , mathematics consists of simple , intuitive axioms that work together to form much more compound axioms known as theorems. A system of mathematics can be thought of as a building consisting of sub-structures (theorems) that further consists of bricks(axioms). The act of laying down each brick or sub-structure corresponds to the act of proving(proofs). The existence of sub-structures simplify the process of building by having already proven theorems at its disposal. As a rule of thumb, the more the number of theorems that already exists inside the system, the easier it is to prove a new theorem.This process continues, expanding the net pool of theorems and axioms inside that system making it stronger. However, the fabric of each mathematical system is extremely sensitive to error. Meaning, a theorem that can be proved to be both true and false, can singlehandledly cause the entire system to explode disproving every single theorem inside that system.
To understand why, let us understand the principle of explosion.
PRINCIPLE OF EXPLOSION
As discussed earlier, each axiom/theorem depends on every other axiom/theorem in some way as we use these already theorems during our proofs. But what happens if we can prove that some axiom inside is system is both correct and incorrect?
Let us consider an axiom A inside system S. If we say that A is both true and false, we are essentially saying that the entire system of mathematics is both true and false i.e. unreliable in practice.
As an example, consider two contradictory statements – “All apples are red” and “Not all apples are red”, and suppose (for the sake of argument) that both are simultaneously true. If that is the case, anything can be proven, e.g. “the sky is blue”, by using the following argument:
We know that “All apples are red” as it is defined to be true.
Therefore, the statement that (“All apples are red” OR “the sky is blue”) must also be true, since the first part is true.
However, if we were to apply “Not all apples are red” which is also true, unicorns must exist – otherwise statement 2 would be false. It has thus been “proven” that the sky is blue. The same could be applied to any assertion, including the statement “the sky is NOT blue”.
If the above part was confusing, the principle is essentially saying that if one axiom is ambiguous/self-contradictory, then everything else is also the same.
GODEL’S INCOMPLETENESS THEOREMS
First theorem :
Any sufficiently expressive system of mathematics is either incomplete or inconsistent.
For the sake of argument, say, our system is incomplete. No matter how many theorems we prove, there will always be new ones ready to be proven. Thus, a system will never be complete. But what if our system is inconsistent? Is there a way to tell that it is so? Let us take a look at the second theorem.
Second theorem:
A consistent math system cannot prove its own consistency.
According to this theorem, we cannot determine whether or not our system is consistent. This is because, in trying to prove so, we can use any of contradictory statements leading to error.
To conclude, mathematics is both incomplete and perhaps, inconsistent, but we’ll never know unless of course, we stumble upon some practical error caused as a result of mathematics.
]]>In faith-based societies, life is governed by fate and magic. Nothing needs to be explained—things happen because of fate, or the will of the Gods, or some other magical explanation.
Therefore, when Socrates says that wisdom is to know that you know nothing, it is a paradigm shift. He is not saying—I don’t know the will of the Gods, or I misinterpreted the oracle, or I am not sure where my fate will lead me; he just says I do not know. In fact, he was tried and executed for corrupting the youth of his day because he stopped people and asked them how they knew what they knew.
This is best explained by Plato’s allegory of a cave in ‘The Republic’. He says that if you were to imagine that prisoners were tied in a cave such that all they could see was the wall opposite them, and they could not see anything else—all they would know of the world through observation would be the wall opposite them.
Now, if someone put puppets in front of a fire behind them and made dancing forms on the wall opposite, they would think that those large, dancing forms were what being were like. But this is vastly different from what the world is really like. Therefore, our knowledge of the world is limited by the limits of our observation and perception.
Imagine what this implies:
Here, it seems to me that Plato is just talking about the limits of observation, and not the veil of biases that surround us that prevent us from observing something without preconceived notions. He says that if the men in the cave were taken outside and allowed to see actual humans and the actual sun, they would not be able to value the knowledge produced inside the cave again. They may even be considered mad once they returned to the cave and reported what they saw.
But I wonder, when the prisoners went outside the cave and saw humans in their actual form—would they recognize that this is the true form of the human, or think that these were mutant versions of actual humans because they did not resemble the shadow forms dancing in the light? [more on this later]
The bias towards observation can also be seen in Greek mathematics. Now, we all know that Pythagoras gave us equations such as:
a^{2} + b^{2 }= c^{2}
for hypotenuse triangles, and the Greeks made major advancements in geometry and mathematics.
But did you know that the equations were not written in this form or in any form we consider mathematics today? The equations were written discursively and proved by actually filling two squares, one with side a and one with side b, into a square of side c (a square of side x is x^{2}). The a square and the b square were actually cut into smaller and smaller squares and fit into the c square. All mathematical proofs by the Greeks were in this form [Read more about this here].
Fig. This is the Greeks calculating the value of pi by fitting smaller and smaller squares in a circle (Source: Heaton, 2015).
Sources:
Complete these sums in NB2 and Show it on 1/05/2017.
Regards
Maulik Trivedi
]]>Complete the remaining questions of Ex. 2.4 in NB2.
]]>Complete the remaining questions of Q3 and 4 of Ex 2.2 in NB2.
]]>… Oh and by the way, I’d recommend giving a try of the first task on Project Euler first, and if you get stuck, have a look back here for some inspiration! Or if you have already completed it, I want to know your method of thinking, it could be a better solution!
The first task I will be completing will be in regards to arithmetic series’, or more direct to the task:
“If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.”
Sounds pretty simple right? Well, one thing I have learned from programming, is the 4 P’s (Planning Prevents poor Performance) and that saying couldn’t be more true. So I went a bit in depth, and take a look at how I went around this problem.
The Formula behind Arithmetic Series is this:
Where the total sum of the integers in a series is found by:
n = Total terms in a series
a1 = First term
an = Last Term
This formula can then be used for working out a “Rough Estimate” for our final answer (Important: You will see why it is only an estimate in a minute!)
n = 1000/3 = 333.3 Therefore 333 terms.
a1 = 3
an = 333 * 3 = 999
Formula: (333(3 + 999)) / 2
= 166,833
n = 1000/5 = 200 Therefore 199 terms (last term would equal 1000, rather than be less than)
a1 = 5
an = 199 * 5 = 995
Formula: (199(5 + 995)) / 2
= 99,500
Approximated Answer = 99,500 + 166,833 = 266, 333
The real answer is slightly different. Now this is because, what happens with common multiples? They aren’t added twice, eg: 15 isn’t added twice due to (5 * 3) and (3 * 5). This means that 15 will only be added once, and thus the real value will be less than 266, 333 (Sum of 5’s and 3’s).
So get some water, write some notes, make a plan, and think of how exciting and fascinating this coding adventure really is (Of course)…
For the development of my program to do this, I first chose a good IDE for a good language. I prefer C# so I’m writing in visual studio, however something like python isn’t that bad either…
Tip: Do some pseudo code first if you are completely stuck, or if you just want to be sure you are heading down the right path. At times throughout my blog, I will be doing Pseudo code, at other times… Naff it.
Answer = 233,168
Overall, I couldn’t have thought of a much more simpler way to do this mathematical puzzle. Once the program had been run, I got an answer… and it was correct! I entered it on Project Euler, and now I’m thinking about my move for the second task!
It also had completed the puzzle under the one minute rule, where if the answer takes more than one minute to generate, your algorithm is inefficient. All methods of completion for all of the challenges are designed to be suitable for this rule.
Got a burning question? Ask it!
Done it differently? Show it!
Hope you enjoyed this,
Alpha. :)
Mathematics is a subject in secondary and sixth form education where the stereotypes of a maths lesson or a maths teaching style is still prominent. What I mean is that many people still associate mathematics with the non-interactive self-contained lessons they used to be. As a student I can fully say maths teaching has changed! For the better at least…
Yes, maths teaching has got better with use of online and computer resources etc. but how is classical maths classes changing. Well, already maths classes have been using more and more real life links between where you use your mathematics in a practical basis. An example would be when teaching younger students the concept of volume within mathematics and how this is demonstrated through practical learning where they discover the formulae and the methods themselves.
This pedagogy of self learning and discovery I found was the best method for myself when learning new content from the syllabus as this way it seemed to develop a deeper understanding of where the mathematics has come from and how its applied to other scenarios.
Maths teaching has changed over many years and will continue to change with the new A Levels in September 2017. Lets see how this change in specifications copes with current mathematical methods of teaching and maybe how the teaching has to adapt…
I hoped you enjoyed the read and feel free to email about the pedagogy (methods of teaching) you have experienced and what has been effective!
Keep Learning!
]]>Yesterday, I went back to take another look. That was mostly to see if there were any additional comments. And there were two, both by Robin Herbert. But comments are now closed for that post. So I’ll say something here.
First some links:
Both comments add to the discussion and are worth reading.
In his first comment, Robin says:
So the argument that fictionalism must be true because the axioms are only conventions appears to make the same mistake as saying the truth or falsity of “if A then B” depends on the truth or falsity of A.
To me, this seems weird. I have said that I am a fictionalist. I have never said that fictionalism is true. I’m not at all sure that I know what it would even mean to say that fictionalism is true.
When I say that I am a fictionalist, what I mean is simply that when I am doing mathematics, I am treating mathematical entities as if they are useful fictions. It is useful to be able to give an existence proof for the solution of an equation. And, in talk about that, it is convenient to be able to say “a solution exists.” But it has always seemed to me that this is not the ordinary real-life meaning of “exist”. It is, instead, a special meaning of “exist” that is used in talk about mathematics. So I can see that as saying that it exists in the fictional world of mathematical entities. I don’t have to require any actual existence beyond that.
If I meet a mathematical platonist, I don’t say to him “your platonism is false”. I’m skeptical of platonism, but I’m not going to criticize a mathematician who holds to a platonist philosophy. It doesn’t really matter to me whether platonism is true or false, or whether fictionalism is true or false. And I doubt that it even makes sense to talk about whether fictionalism or platonism is true or false.
In his second comment, Robin writes:
I started as a Fictionalist. If you asked me to define mathematics I would have said that is easy, mathematics consists of choosing a set of symbols and then making up some rules to manipulate them by and then proceeding to manipulate them by those rules.
Well, that’s weird, too. What Robin is defining as “fictionalism” is what I would refer to as “formalism”. And I see formalism as quite distinct from fictionalism.
If I say
I am not thinking of “3”, “4” and “7” as symbols to which I am applying rules. Rather, I am thinking of “3”, “4” and “7” as names of counts, and in thinking about that arithmetic expression I am thinking about counts rather than about symbols. But nothing was actually counted. So they are names of fictional counts. And that’s why I think of myself as a fictionalist. When I say “3 + 4 = 7”, I am really saying something about what should ideally be the consequences of counting behavior. So my fictionalism is related to my behaviorism.
My understanding is that a platonist would say that “3”, “4” and “7” actually exist in a platonic world of mathematical entities, and that counting selects an entity from that platonic world.
According to Robin’s comments, he was once a fictionalist. But that did not work out for him and he is now a platonist. But Robin explains how he thinks of mathematics. And, on reading that explanation, it seems to me that Robin actually is a fictionalist and not a platonist.
When he says that he was once a fictionalist, but changed to a platonist, I am reading that as “he was once a formalist, but changed to being a fictionalist”.
In his first comment, Robin expresses an “if … then” view. Everything we claim to be true about mathematics is an “if … then” kind of truth. That is, if we assume the axioms to be true, then it follows that the theorems are true. My view is similar. But platonists don’t talk that way. The platonists with whom I have discussed mathematics, seem to believe that there is an actual truth to be found. And if we cannot prove it from our axioms, then maybe we have the wrong axioms. They tend to see mathematics as a science of the platonist world, with axioms as analogues of the scientific laws.
Clearly, Robin and I disagree on the meaning of “fictionalism” and “platonism”. Just as clearly this is not a fighting disagreement. We can disagree without being disagreeable about it.
I see this as just one illustration of why I take meaning to be subjective. That seems to disagree with the view of most philosophers. Maybe I’ll come up with a separate post on this issue.
]]>It’s a little-known fact that rhombicosidodecahedra prefer to fly in flocks of sixty, as seen here. I made this using Stella 4d, available at this website.
]]>And yet, if you ask an educated person “the value question”:
What does one dollar, or one pound, measure or represent?
then you are likely to be met with a good few minutes of rambling and mumbling.
Everyone knows that the marks on a ruler measure distance, or a thermometer’s mercury column measures temperature, or a clock’s hands represent time. And inquisitive minds, before they are socialised to stop worrying about such things, naturally ask the value question and enquire about the nature of the numbers they find stamped upon the goods they buy, and the tokens they carry in their pockets. But unlike rulers, thermometers or clocks, few adults have a clear and distinct idea of the semantics of monetary phenomena, including economists.
Possible answers to the value question include “some specific thing”, “many things” or “nothing”. The history of economic thought has explored all these options.
However, the predominant attitude among economists today is value nihilism. “There is only price” and to seek something behind prices, to dig deeper, is simply a kind of confused essentialism. In consequence, to ask a modern economist the value question is akin to raising the issue of phlogiston with a modern physicist. It is anachronistic. Economic science once grappled with the value question but has subsequently educated itself to stop asking it.
The academy, at least within capitalist societies, turned against the classical labour theory of value during the 19th Century’s marginal revolution in economics. Subsequently, the labour theory has eked out a threadbare existence on the periphery of the academy, while enjoying robust and continued support from a small minority of intellectuals associated with the socialist tradition within civil society.
But even a resolutely pro capitalist academy, like we experience today, must appear to conform to scientific norms. So what reasons are normally given for rejecting the labour theory of value?
Simplifying, the academy normally offers two main reasons: one exoteric, and the other esoteric.
The exoteric reason is that market prices are determined by the Marshallian scissors of supply and demand. So prices are indices of scarcity, and therefore cannot represent the amount of labour time supplied to produce commodities. This kind of argument frequently appears in popular or “folk” rejections of the labour theory of value.
The esoteric reason is that Marx’s theory of the transformation doesn’t work. What is that theory? Marx understood that equilibrium (as opposed to market) prices of commodities systematically diverge from the labour time supplied to produce them. So the labour theory of value appears false on the empirical surface of capitalist society. Yet Marx argued that this divergence is merely apparent and caused by the distorting effect of capitalist property relations. In his unfinished notes, published as Volume III of Capital, he proposed that prices are conservative transforms of labour time (i.e., prices are “transformed” expressions of labour time). So although the prices of individual commodities and labour times diverge, there is a one-to-one relationship between prices and labour times in the aggregate.
The father of neo-classical economics, Paul Samuelson, published articles in the 1950s and 70s that, although not original, demonstrated in mathematical terms that Marx’s theory of the transformation cannot work, and therefore there isn’t a systematic one-to-one relationship between equilibrium prices and labour time.
Unsophisticated critics of the labour theory will offer the exoteric reason, but more sophisticated critics know the scarcity objection doesn’t hold water. So sophisticated critics ultimately defer to Marx’s transformation problem.
But here’s the rub. The critics have a point. Marx’s theory of the transformation is indeed incomplete and does have its problems — a feature that Marx first pointed out himself in his own notes.
From a sociological point of view, and simplifying greatly, we have, on the one hand, a pro capitalist academy eager to excise the labour theory of value, and all its radical implications, from academic discourse; and, on the other hand, a pro Marxist periphery motivated to defend the theory against the ideological attacks of the ruling elites. The environment for pursuing the science of the theory of value is decidedly unhealthy. But it could not be otherwise.
An unfortunate trend in Marxist circles, which represents a real obstacle to material progress, is to “wiggle out” of the transformation problem via creative reinterpretation of the meaning of Marx’s texts. Many reinterpretations attempt to save Marx only by dismantling the scientific content of Capital. For example, a large family of reinterpretations end up denying that labour time is a market-independent property of reality. So much for materialism.
So why is the labour theory of value true? I give a brief, technical answer in this new position paper:
The General Theory of Labour Value
Some of the main points are:
The modern nihilist attitude does not represent a sophisticated rejection of naive substance theories of value but instead signifies the continued existence of unresolved and fundamental theoretical problems that first manifested at the birth of modern economic science in the eighteenth and nineteenth centuries.
There is no royal road to science. And the unfortunate truth for the pro capitalist academy is that the road to a scientific understanding of the economy passes through Marx. There’s no way around him, because, of all the economic thinkers, he got the fundamentals of the theory of value right. And every modern school of economic thought, whether orthodox or heterodox, is woefully ignorant of what the unit of account actually signifies. We are like blind ants who obsess, suck and exchange the Queen substance, yet know nothing of its true function. Marx stands on the road ahead, pointing in the direction of a truly scientific understanding of the kind of society we live in. That’s why this post has a picture of a coin with Marx’s head on it.
]]>Also consider these questions:
What happens if you are multiplying or dividing a fraction by a whole number?
What if you are calculating with mixed numbers? What would you do first?
What learning questions do you still have?
Explore this website to support you with methods and strategies:
http://www.mathsisfun.com/fractions_division.html
Happy Maths! :-)
]]>Here are some of the photos with their accompanying questions:
Beyblades:
Hair:
Birthday Balloons:
From Problem Solving to Problem Posing
What is the purpose of getting students to write mathematical problems? First of all, the problems give us good insight into whether students recognise mathematical situations, and whether they understand where, how, and what mathematics is applied in day to day situations. An added bonus is that the students are highly engaged because they have ownership of the mathematics they are generating, the topics they choose are of interest to them, and stereotypical perceptions of school mathematics are disrupted.
Student Reflection
The students who wrote the examples above completed a structured written reflection following the sequence of designing and solving each others’ maths curses. Here are some of reflection prompts and a sample of responses:
What did you enjoy about today’s learning?
“working with my team”
“working at the problems for a long time and then finally getting them after a long, hard discussion”
“solving questions that my friends wrote”
“I felt challenged and I learnt more about what maths is”
“working with my group, choosing our own questions and learning something new”
“I liked the chess card the best because we had to solve it together and use problem solving”
“having a go at tricky questions even if i got them wrong”
Did you learn anything new?
“how to work things out in different ways”
“working in groups helps you learn more skills”
“not every question uses just one skill like addition, division, multiplication or subtraction”
“when I am challenged I learn more”
“Maths is not always easy”
“how to work together”
“Everyone in the group has different responses so we needed proof to figure out the right one”
What surprised you about this task?
“It surprised me how hard my own questions were”
“I didn’t know that we could come up with so many interesting questions”
“I got a shock! We had to research to solve some problems, Adam even taught me how to add a different way”
“I got some questions wrong “
“It was hard but if we put our brains into gear we could figure it out”
“I was able to play while doing maths”
Using activities such as this provides multiple benefits for students. Contextualising the mathematics using students’ interests highlights the relevance of the curriculum, improves student engagement, and makes mathematics meaningful, fun and engaging!
]]>Write a quadratic polynomial, the sum and product of whose zeroes are 3 and -2.
From a resource by Ross, Keppler, Canil & O’Neill.
]]>
Complete Q2 of Ex 2.2 of NCERT TB in NB2.
]]>