onsdagen den 30:e september 2009
I have just upon invitation submitted the article The Mathematical Secret of Flight to Normat Nordisk Matematisk Tidskrift.
The article explains in easily understandable form how a wing generates lift at the expense of small drag, based on a stability analysis of potential flow and computations of turbulent solutions of the incompressible Navier-Stokes equations for slightly viscous flow. It is also shown that the state-of-the-art explanation of lift by Kutta-Zhukovsky based on connecting lift to circulation, is physically incorrect.
The new explanation of the mathematics and physics of flight is also presented in the knol
Why It Is Possible to Fly including more references to my work with Johan Hoffman.
The book Simulating Minds, The Philosophy, Psychology, and Neuroscience of Mindreading, by Alvin I. Goldman discusses aspects of the probably unique capability of Homo Sapiens among animal species to mindread or simulate other peoples minds in social interaction. Minds reading other minds is what is going on all day long when we play our social and professional games.
and the blog post Reality of the Virtual and Virtual Reality.
What would a modern engineering education boosted by computational mathematics look like? Let us see:
Mathematical models of physical systems consist of system of partial differential equations of the form:
(pde) du/dt = f(u,Du,x,t)
where u(x,t) is a given function of a space-time coordinate (x,t) and f(u,Du,x,t) is a function of
(u,Du,x,t) where Du represents derivatives of u with respect to x. The partial differential equation or pde is supposed to hold for x in some domain in space, and t in some interval of time, and the pde is complemented by certain given boundary conditions at the boundary of the domain and a given value of u at initial time. The idea is to determine the function u(x,t) satisfying (pde) for some given data, because the function u(x,t) carries information about the output response of the system to the given input data.
As a basic example: Newton's equations of motion take the form
dX/dt = V, mdV/dt = F(X,V,t)
where X(t) and V(t) denote the position and velocity at time t of a pointlike body of mass m acted upon by a force F. Solving this equation with given initial position and velocity will give
information about these quantities at a later time. Genial! And very useful in enginering design
and control. If you want to direct a rocket to hit Mars for example, you can to that by solving Newton's equations, if you only know where Mars is.
Once you have formulated the pde with given data, you use a computer to compute the solution
using the finite element method. This leads to the following plan for your engineering studies: Learn how to
- formulate engineering problems as pdes using calculus and linear algebra
- solve pdes using a calculus and linear algebra boosted by a computer
- determine input data.
This will take you about a year, and will bring to the level of present education after 4 years.
The remaining 3 years of a 4 year education, you can spend by digging deeper into
whatever field you may interested in and inventing new devices designed by computation.
What do you say? Would you be interested in such an education if was available?
The proposed program in Simulation Technology at KTH could fill your need. The trouble is that it has not yet started, because traditional program administrators are not enthousiastic about a competing modern program. As a student you can help to balance forces by expressing your interest in the new program.
tisdagen den 29:e september 2009
Different fields of engineering concern different phenomena described by different often named differential equations as described in How to Not Organize a University including
- mechanical: Navier's eqns for solids and Navier-Stokes eqns for fluids
- electrical: Maxwell's eqns for electromagnetism
- chemical: Reaction-convection-diffusion eqns
- quantum physics: Schrödinger's eqn
- classical physics: Lagrange's eqn
- structural: beam, plate and shell eqns.
Traditional engineering is based on various special methods for solving these equations using analytical mathematics, handbook formulas, tables and calculations by slide ruler or mechanical calculator. Each field has developed it own special bag of tricks for this purpose,
and this bag of tricks forms the core of engineering education in each field.
The 68 finite element revolution has brought a fundamental change: All of the different differential equations can be solved by one and the same methodology! This has changed engineering practice and is now opening to more effective education: Instead of spending years to learn a special bag of tricks, which no longer is used in practice, in a special domain of engineering, the student can learn a general methodology opening to interdisciplinary studies and practice.
The modern engineer equipped with a modern general computational methodology for simulation will be able to combine knowledge from different fields of engineering in new forms of interdisciplinary work, and this is what society needs...
Engineering education today largely is the same as 50 years ago, and the 68 computational revolution has not yet changed much. But the wave is building up and changes are around the corner...stay tuned...on the surfing board towards modern engineering studies...
måndagen den 28:e september 2009
In the Swedish debate on climate change, critique of climate alarmism is often described as pseudo-science. With this arguement questioning of the reliability of climate models can be dismissed as pseudo-science.
But isn't it the other way around? Isn't it necessary to show that climate models are reliable in order for climate modeling to be anything but pseudo-science?
This is the first post in a series directed to science and engineering students with the hope of encouraging to independent search for knowledge and know-how using the tools of modern information society.
The traditional sign of an engineer is the slide ruler showing that the job of an engineer is to compute. To build a skyscraper, bridge, car, airplane, boat, TV..., an engineer first has to make a design and by computation secure that it will work. Trial and error when building a bridge is not feasible.
For that purpose the design is modeled mathematically typically as systems of differential equations expressing basic laws of physics and mechanics, and the equations then have to be solved with some input data to obtain a solution carrying information of interest for the design, like the force in the cable of a cable-bridge, or engine power required for take-off of a jumbojet.
The civil engineering education I started in 1963 was based on
- analytical computations with mathematical symbols by pencil and paper
- numerical computations by slide ruler or mechanical calculator
- tables and diagrams of solutions of basic differential equations
- handbook formulas.
In the basic subject of structural mechanics I was taught how to use these tools to design
basic construction elements like beams and constructions assembled from basic elements like the frame of a building or a bridge.
However, at the end of my education in 1968 my basic training had become obsolete: A revolution had occurred, the famous 68 revolution introducing the new era of the finite element method: The idea was simple and powerful: Solve the differential equations using a computer instead of tables and slide ruler. This was done by decomposing the structure to be analyzed into many small elements with known simple behavior, which the computer could assemble in computational simulation of the assembly.
Genial! By the finite element method you can solve any differential equation in any field of engineering! This puts a turbo to mathematics, physics and mechanics in the hands of the able engineer. This is how weather predictions are made, for example, as well as the simulations of global climate hiding the future of mankind in its digits.
As an engineering student of course you expect that you will learn to use this powerful tool during your basic education, right? But no! Engineering education has not changed much
since 1963 and the 68 revolution has had no impact on the basic education in mathematics and mechanics! It is the same today as 50 years ago in 1959, the year Ingemar Johansson took the heavy weight world title from Floyd Paterson, which people like me still remember!
Yes, there is a change since 1963: The handbook and slide ruler are no longer used, but the computer with its finite elements is still waiting to get introduced in basic training. The result is an engineering education which no longer teaches the engineering skills needed in research and industry. The result is an eroded empty education which no longer attracts the students. The result is decreasing interest from school teachers and high school students in mathematics and science.
If you anyway want to pursue an engineering education, you can balance the traditional education you will meet by opening a window to the new world of computer simulation or simulation technology with the double meaning of
- technology for simulation: how make computer simulations
- technology with simulation: how to learn and use technology by computer simulation.
You can do that by browsing My Book of Knols as a general introduction to science and mathematics, starting with
then taking a look at
and ending your first encounter by understanding with the help of computational simulation
This is a real showcase for computational simulation, since traditional science missed the true mechanism of flight. Read and think and do not accept as a truth everything that is written in a book, in particular not if it is a flight mechanics book!
I will soon be back with more suggestions how to learn about the new world of simulation of 68... Students aiming at KTH may be interested in the proposed new program in Simulation Technology.
In our new information society each student must take an active part in the education already from start, to search information and to learn. This is of particular importance in engineering education since it is so old-fashioned. The massproduced slide ruler engineer is dead, but the new age of simulation has not yet entered basic education, and so you as a student will have to take responsability for a rebirth of the engineer...the computing simulating multidisciplinary engineer of the information society...Is this clear?
lördagen den 26:e september 2009
As a preparation to the UN Convention on Climate Change in Copenhagen in December the International Alliance of Research Universities organised an international scientific congress on climate change, Climate Change: Global Risks, Challenges and Decisions, in Copenhagen March 10-12, which produced a Synthesis Report as an update of the IPCC AR4 with the following key messages:
- Recent observations show that greenhouse gas emissions and many aspects of the climate are changing near the upper boundary of the IPCC range of projections.
- Many key climate indicators are already moving beyond the patterns of natural variability within which contemporary society and economy have developed and thrived.
- These indicators include global mean surface temperature, sea-level rise, global ocean temperature, Arctic sea ice extent, ocean acidification, and extreme climatic events. With unabated emissions, many trends in climate will likely accelerate, leading to an increasing risk of abrupt or irreversible climatic shifts.
- Temperature rises above 2 degrees Celsius will be difficult for contemporary societies to cope with, and are likely to cause major societal and environmental disruptions through the rest of the century and beyond.
- Rapid, sustained, and effective mitigation based on coordinated global and regional action is required to avoid “dangerous climate change” regardless of how it is defined. Weaker targets for 2020 increase the risk of serious impacts, including the crossing of tipping points, and make the task of meeting 2050 targets more difficult and costly.
- Setting a credible long-term price for carbon and the adoption of policies that promote energy efficiency and low-carbon technologies are central to effective mitigation.
- Climate change is having, and will have, strongly differential effects on people within and between countries and regions, on this generation and future generations, and on human societies and the natural world.
- Tackling climate change should be seen as integral to the broader goals of enhancing socioeconomic development and equity throughout the world.
- Society already has many tools and approaches – economic, technological, behavioural, and managerial – to deal effectively with the climate change challenge. If these tools are not vigorously and widely implemented, adaptation to the unavoidable climate change and the societal transformation required to decarbonise economies will not be achieved.
- A wide range of benefits will flow from a concerted effort to achieve effective and rapid adaptation and mitigation. These include job growth in the sustainable energy sector; reductions in the health, social, economic and environmental costs of climate change; and the repair of ecosystems and revitalisation of ecosystem services.
- If the societal transformation required to meet the climate change challenge is to be achieved, then a number of significant constraints must be overcome and critical opportunities seized. These include reducing inertia in social and economic systems; building on a growing public desire for governments to act on climate change; reducing activities that increase greenhouse gas emissions and reduce resilience (e.g. subsidies); and enabling the shifts from ineffective governance and weak institutions to innovative leadership in government, the private sector and civil society.
- Linking climate change with broader sustainable consumption and production concerns, human rights issues and democratic values is crucial for shifting societies towards more sustainable development pathways.
Here the 2 degree limit is formulated, which will dominate the Copenhagen meeting, as well as the need to reduce inertia in social and economic systems and the idea of linking climate change to human rights issues and democratic values. The background consists of vague but suggestive catastrophy scenarios:
- many aspects of the climate are changing near the upper boundary of the IPCC range of projections
- many key climate indicators are already moving beyond the patterns of natural variability
- major societal and environmental disruptions through the rest of the century and beyond
- crossing of tipping points.
The Synthesis Report is clearly politicised with science subordinate to the higher goals to decarbonise economies and create job growth in the sustainable energy sector.
torsdagen den 24:e september 2009
The question of short-time vs long-time accuracy of climate models is discussed in a recent article in New York Times by Andrew Revkin. What does the plateau in global temperature seen over the last decade tell us? That global warming is over? Maybe, maybe not. But it tells that climate models are inaccurate over a decade since the plateau was not predicted. Does it then follow that climate models are accurate over long-time over a century? Not necessarily. A short-time inaccurate model may be long-time inaccurate as well.
What is then the mathematical nature of models which are short-time inaccurate but long-time accurate. Are there such models? Yes, if you put a rapid short-time oscillation on top of a long-time slow motion, then a model that discards the rapid short-time oscillation will have this property.
Can you then draw the conclusion that a simple model with some form of damping of short-time variations, will be long-time accurate? Sometimes yes, but then only under very special conditions: regular short-time oscillation and linearity, which allows a lot of cancellation to take place. But in a non-linear system this may not be the case at all, since cancellation maybe lost.
The idea that climate models can be allowed to be short-time inaccurate, without jeopardizing long-time accuracy, thus seems questionable. Of course, a climate model will be short-time inaccurate if some unknown major natural variation is not taken into account, and if that unknown natural variation would disappear, then the model may recover long-time accuracy but then together with short-time accuracy. The combination of short-time-inaccurate with long-time-accurate seems very special.
Altogether, it seems reasonable to ask climate models to be short-time accurate. If they are not, it probably signals that they are also long-time inaccurate. Compare the recent post by Roger Pielke Sr. stating in particular:
- Societally useful (i.e., reliable, accurate, etc.) climate prediction requires that all of the feedbacks and other physical processes included in weather prediction be presented in the climate prediction model. In addition, longer-term feedback and physical processes must be included. This makes climate prediction amuch more difficult problem than weather prediction.
We learn that climate models also have to be weather models, and thus have to be short-time accurate in order to have a chance to be long-time accurate. It follows that the accuracy of
climate models can be tested over short-time. What is the result of such tests?
I will participate in a Seminar on New Perspectives in Teaching Mathematics at Helsinki University of Technology, Oct 26, described as
- Teaching mathematics is always a challenging task. Recent advances in didactics and technology offer new opportunities in advanced mathematics teaching. A number of national and international experts share their views in a seminar organised by the Department of Mathematics and Systems Analysis, TKK. The purpose of the seminar is to discuss the implementation of new teaching methods at technical universities.
The debate about mathematics education reform is picking up...in Finland...if not in Sweden...but stay tuned...
The title of my talk, with an outline here, is:
- Mathematics Teaching of the IT-Society: What, Why and How?
The following Knol articles and blog posts give an introduction and background to the talk:
- Traditional Role of Mathematics Education in Society??
- Why/What Mathematics for Engineers
- Modern Mathematics Education
- Why Teach Mathematics?
- Simulation Technology
- The Body&Soul Project home page
- How to Not Organize a University
- Swedish Mathematics Flu
- The Unreasonable Effectiveness of Mathematics Education
- The Reasonable Ineffectiveness of Mathematics
- Wigner Without Computer Unreasonable
- Mathematics and IT
- Mathematics Integrated with IT in School Education?
- Computer Games: Learning with Feed-Back
- The Ideology of Mathematics
- A Critical Analysis of the Ideology of Mathematics
- Mathematics and Religion vs Self-Help
- Analysis of Presentation of Mathematics Courses at HUT.
For more material on various themes, see My Book of Knols.
tisdagen den 22:e september 2009
President Obama stated in his UN speech on climate yesterday:
- Rising sea levels threaten every coastline. More powerful storms and floods threaten every continent. More frequent drought and crop failures breed hunger and conflict in places where hunger and conflict already thrive. On shrinking islands, families are already being forced to flee their homes as climate refugees.
- As John F. Kennedy once observed: “Our problems are man-made, therefore they may be solved by man.”
- But this is a new day. It is a new era. And I am proud to say that the United States has done more to promote clean energy and reduce carbon pollution in the last eight months than at any other time in our history.
Obama knows very well that he cannot deliver any US reductions to Copenhagen in December, but compensates by instead expanding the goal:
- What we are seeking, after all, is not simply an agreement to limit greenhouse gas emissions. We seek an agreement that will allow all nations to grow and raise living standards without endangering the planet.
Obama knows that "deniers" dominate the US Senate and according to a recent Blomberg poll only 2% of the US voters believe climate change is the most important issue facing the country, but compensates by
- The good news is that after too many years of inaction and denial, there is finally widespread recognition of the urgency of the challenge before us. We also cannot allow the old divisions that have characterized the climate debate for so many years to block our progress.
Obama's message to the developing world is:
- But those rapidly-growing developing nations that will produce nearly all the growth in global carbon emissions in the decades ahead must do their part as well. It will do little good to alleviate poverty if you can no longer harvest your crops or find drinkable water,
which can be read as letting emission cut come before alleviating poverty. Obama repeats the mantra:
- We know that our planet’s future depends on a global commitment to permanently reduce greenhouse gas pollution,
but must know that this is precisely what scientists cannot agree to know. Convincing?
Measured data tell a different story than Obama does. Check yourself!
Olle Häggström frågar på Uppsalainitiativets blogg om Klimatskeptikerna behöver bemötas?och besvarar sin fråga på följande sätt:
- Politiker i USA som faktiskt anammat det klimatskeptiska budskapet är de inte så få - och inte heller maktlösa - just nu verkar det dessvärre som om de lyckats sabotera Obamas ambitioner inför klimatmötet i Köpenhamn.
- I Sverige är läget, precis som när det gäller konflikten mellan evolutionsbiologi och kreationism, betydligt ljusare (mer upplyst!) än i USA. I Riksdagen har vi bara enstaka klimatskeptiker.
- Men hur kan vi känna oss säkra på att dessa pseudovetenskapliga strömningar inte breder ut sig ytterligare? Nu visar det ju sig att vi inte behöver gå längre än till Danmark för att konstatera att ett parti som i senaste folketingsvalet fick mer än 13% av väljarstödet (och som i själva verket tillskansat sig en position där de med sitt främlingsfientliga budskap i hög grad sätter hela landets politiska agenda) går ut med ett öppet klimatskeptiskt budskap.
Bravo Olle: Att associera kritik av klimatalarmism med kreationism, främlingsfientlighet och pseudovetenskap samt krydda med lite USA-hat, är väl avsett som en uppmjukning av kritiken inför mötet Att bemöta desinformation i klimatfrågan.
Men handen på hjärtat Olle: Var finns vetenskapen i detta Ditt budskap? Är det inte Du som levererar desinformation? Och varför gör Du det, som vetenskapsman?
The goal for the Climate Change Conference in Copenhagen of a global agreement on reduction of CO2 emissions is now being turned down into instead an agreement to limit global warming to 2 degrees Celcius. Swedish Minister of Environment Andreas Carlgren leading EU at the conference expresses his ambition as follows:
- The is key is to continue to keep the 2 degree limit within reach.
- If we focus on the 2 degree limit, then we can put aside discussions on emission control.
What a clever idea! But why limit this clever limit idea to only global warming? Why not agree to limit
- global cooling to - 2 degrees and avoid the coming ice age?
- global economy growth to minimum 2% and avoid economic downturn?
- the speed of light to 200.000 km/s?
- budget deficits to 2%?
- bank loan losses in the Baltic states to 2%?
Of course without any discussion of how this may be accomplished. But do voters really swallow such arguments? Or only politicians?
måndagen den 21:e september 2009
The main argument of climate alarmism supported by IPCC is that ice core records show a close correlation between the concentration of CO2 in the atmosphere and global temperature over the last four glacial-interglacial cycles. This is taken as evidence that CO2 is an important greenhouse gas, which introduced in climate models predicts that doubling of preindustrial CO2 could cause global warming of up to 6 degrees Celcius. On this basis IPCC recommends control of CO2 emissions, to be negotiated at the UN Climate Change Conference in Copenhagen in December as a continuation of the Kyoto Protocol.
However, the data shows that CO2 lags temperature with about 800 years, which is compatible with the fact that a warm ocean releases more CO2, just as a warm bottle of coke.
The evidence thus suggests that it is the temperature that influences the CO2 concentration in the atmosphere, and not the other way around. The fundamental argument of climate alarmism supplied by IPCC thus seems to lack rationale...or maybe the dependence is not as simple like that...see the discussion below...anyway the fundamental scientific problem is this: What is the climate sensitivity or the effect on global temperature of carbondioxide emission?
DNs huvudledare 18.9 innehåller under temat Fem punkter för kunskap följande avsnitt rubricerat Se Matematiken:
- Matematiken har som kärnämne en formellt gynnad ställning i skolan. Utan godkänt i matematik är man inte behörig att läsa vidare på gymnasienivå.
- Men i vardagen är matematiken misshandlad och missförstådd. Det finns underskott på behöriga lärare. Även bland dem som har ämnesutbildning finns många med bristfälliga kunskaper och svalt intresse för ämnet.
- Utvärderingar av svenska elevers matematikkunskaper visar att det finns mönster i deras misstag. Eleverna har ofta nött in metoderna men missförstått dem och tillämpat dem i fel sammanhang.
- Japan är inte längre det framgångsexempel som det uppfattades som för 20 år sedan. Men inom matematikundervisningen förblir landet en förebild.
- I japanska klassrum behandlas inte matematiken som en samling metoder att nöta in utan som ett sätt att tänka och en värld att upptäcka. Det spännande är inte det som står i facit utan att det finns olika vägar till målet.
Hur har ledaravdelningen fått ihop detta virrvar av halva meningar? Inte genom eget tänkande och frågan är då vem som ligger bakom? Vad som spelas upp är misslyckande, sårad stolthet skuldbeläggning samt oklar vision:
- Formellt gynnad ställning: men inte reellt gynnad då?
- Misshandlad och missförstådd i vardagen: på vad sätt och av vem?
- Lärare med svalt intresse och bristfälliga kunskaper lär ut mönster av missförstånd.
- Japan är inte ett framgångsexempel men dock förebild i matematikundervisning.
Det är lätt att genomskåda andemeningen:
- Mer Pengar till matematiker vid universitet.
- Mer Pengar till matematiklärarutbildare vid lärarhögskolor.
Uppenbarligen ett beställningsjobb från Svenska Matematikersamfundet och Nationellt Centrum för Matematikutbildning bakom Matematikdelegationen! Och argumentationen genererar pengar: Jan Björklund satsar 500 milj på att rätta "systematiska fel" eller "mönster av missförstånd" i matematikundervisningen, styrd av propagandafloskler.
Men varför frågar inte DN om detta är vettigt istället för att okritiskt stödja ett sådant slöseri med skattemedel? Om ledaravdelningen anser sig ha en funktion i dagens Sverige, så kan den visa detta genom att öppna för en debatt om matematik, matematiklärarutbildning och matematikundervisning.
Detta är mycket viktigt för Sverige som kunskapsnation! Nuvarande matematikutbildning fungerar uppenbarligen inte och något måste göras. Frågan är vad? Blir det någon debatt på DN? Eller bara okritiska ledare av okunniga ledarskribenter? För att minska okunskapen rekommenderas mina poster om mathematics education och mina knols om mathematics/science education.
onsdagen den 16:e september 2009
A new Oxfam report has today warned that at least 4.5 million children could die unless world leaders deliver additional funds to help poor countries fight the growing impact of climate change, rather than diverting it from existing aid promises.
Rich countries must not steal money from poor hospitals and schools in order to pay their climate debt to the developing world, said Jeremy Hobbs, Executive Director of Oxfam International. But is this realistic?
The idea to strain the already strained economies of poor countries by climate debts, is based on certain alarming predictions obtained by certain computational mathematical models. Suddenly computational mathematics is a question of life and death. If the mathematics alarm is a false alarm, it risks to have meaningless terrible consequences. Who knows if the math is correct? See my posts on climate simulation. All of this is mind-boggling, for a mathematican and human being. Is God Mathematician or is the Devil?
The new Bachelor program in Simulation Technology ST at KTH proposed by the School of Computer Science and Communication CSC, has been approved by the Board of KTH
but the start will be delayed to 2011 or later, according to Dean of Faculty Folke Snickars. The reason seems to be that the School of Engineering Sciences opposes the new program.
At the brink of success, the air goes out of the CSC-ST balloon, or at least out of me. ST is based on a new interactive web-version of Body&Soul , which I intended to supply to KTH. But at my age when every year is a fortune, delays are hard to accept and I thus may have to seek another platform for Body&Soul than CSC-KTH, and there are many possibilities:
Mathematics education is in a deepening crisis, much deeper than the financial crisis, and there is a desperate need of a reformed math education that can bring back students and encourage students to learn useful mathematics. Body&Soul seeks to fill this need and no alternatives are visible: The math department at KTH is representative in offering the same courses as 50 years ago. The time scale at KTH seems to be a century, while the time scale in the World around of today is a year, and this is also my timescale...it is now, or never...for me...
KTH has now decided to start ST at in 2011.
måndagen den 14:e september 2009
Prime Minister Vladimir Putin said on Friday he would reject any new climate change agreement that imposed restrictions on Russia but not on other big polluters, such as the United States or China, Reuters reports.
Russia, currently the world’s third largest polluter behind China and the United States, has offered to cut its greenhouse gas emissions by 10-15 percent below 1990 levels by 2020. However, the country’s emissions were already 34 percent below 1990 levels in 2007, the latest year data were available.
It will be a tough job for Reinfeldt and Carlgren to get together an agreement at the Copenhagen Climate meeting...or maybe not if no cuts are really needed...
At any rate Putin can help solve the climate crisis by a cut of the delivery of gas to EU...
CJ: Can you shortly describe your view on the science of climate change?
PS: Reply is on the way...
The idea of scientific consensus is used by the alarmists of global warming and by the wikipedians controling the information on Wikipedia. However, scientific truth cannot be determined by majority voting, only by scientific facts and arguments. One fact or argument by one scientist can outweigh the consensus of billions of people. Scientific consensus can lead astray, since it can give the false impression of scientific truth, when it is only the superstition of many.
Political truth in democratic societies is determined by consensus of the majority, but scientific truth should not be determined by consensus, in particular not by third-party majority consensus, but by real combat between active living scientists.
It is the same in sports: The Wimbledon matches between Borg and McEnroe were not determined by consensus of the spectators, but by Borg and McEnroe alone.
Borg and McEnroe represent the active living scientists carrying the scientific knowledge at any given time, who by playing matches of science or disputations in seminars and journals decide the current truths.
In sports, you win by walkover if your opponent does not show up to the match, and it is the same in science. You cannot defend your position by saying nothing, neither can the songbird defend its territory by singing nothing.
An illustration is given by the match about how to resolve d'Alembert's paradox in fluid mechanics which has been going on for 255 years. This long match has now come to an end by the publication of my resolution together with Johan Hoffman in the leading Journal of Mathematical Fluid Mechanics. The victory is declared by Google putting our resolution in top position in a search on "resolution of d'Alembert's paradox". It is a walkover victory because the entire editorial board of the Journal of Fluid Mechanics says nothing.
d'Alembert's paradox of zero drag in inviscid flow is important since much of modern fluid mechanics is related to the paradox in one way or the other, as explained in my knols on fluid mechanics.
Michel Crichton expresses the essence very clearly:
- Let's be clear: the work of science has nothing whatever to do with consensus. Consensus is the business of politics. Science, on the contrary, requires only one investigator who happens to be right, which means that he or she has results that are verifiable by reference to the real world. In science, consensus is irrelevant. What is relevant is reproducible results. The greatest scientists in history are great precisely because they broke with the consensus.
- The work of science has nothing whatsoever to do with consensus. There is no such thing as consensus science. If it is consensus, it isn't science. If it's science, it is not consensus. Period."
söndagen den 13:e september 2009
- the basic equations of string theory are not known
- and besides are so incredibly complicated that nobody can understand them even if they were known
- no predictions come out from the unknown equations
- progression has stopped
- physics departments have stopped hiring string theorists
- last hope is that LHC will give new input but LHC does not work...
As concerns the role of mathematics in string theory Voit informs us:
- the mathematics is so sophisticated that only Witten can understand
- but the problem is not so much sophisticated mathematics that nobody understands
- it is that the physical idea of vibrating strings in 10 dimensions does not seem to work out
- physics departments now are saturated with sophisticated math and look for physics instead and string theory moves to math departments.
The idea to integrate mathematics into IT education discussed in a previous post, can really it be seriously considered? Yes, this very natural because
- a laptop is an ideal laboratory for arithmetics, geometry and calculus
- in a couple of all students from early age will use a laptop/mobile extensively
- computer software is based on mathematics
- a laptop gives feed-back
- programming teaches logic
- programming can teach computing
- programming can teach mathematical modeling
- programming can teach problem solving
What more would you like math education to contain? The more I think about this idea, the better it seems. In fact this is the basic idea of Body&Soul.
The recent summit in South Africa co-chaired by President Jakob Zuma and Swedish Prime Minister Fredrik Reinfeldt, who holds the 27-country EU's rotating presidency, dealt with climate change.
Reinfeldt was received with red carpet, military parade and music corps during his first visit to South Africa.
Zuma expressed the view that African countries would veto any climate change deal if rich countries do not meet their demand for money, which some experts said could be up to $200-billion a year.
Reinfeldt expressed the view that developing countries should focus more on the longer-term climate threat than on an economic downturn.
Emission control and poverty in developing countries seems to be message from EU and Reinfelt, which is not acceptable to Zuma and African countries. What does mathematics tell? What will be the deal in Copenhagen, between the rich and the poor?
If Reinfeldt cannot convince South Africans to stay poor or the EU voters to pay the bill, which seems pretty obvious, will he still insist at the prize of loosing his job, or will he rethink, listen to facts of science and focus on something more constructive?
Reinfeldt has shown that he is a pragmatist by reforming the old conservative party into the new moderates replacing the socialdemocrats as the party for everybody, and thus seems to follow reason rather than ideology.
fredagen den 11:e september 2009
Feedback is most essential in learning. A child learns to speak around the age of two with the help of massive parent feedback. A child cannot learn to speak by listening to a record player, because the communication is one-way without feedback.
Traditional school education is largely one-way with little feedback and accordingly is inefficient. On the other hand, our modern IT society offers a flood of feed-back on the web, which is attractive to young minds. Computer games offer so exciting feed-back that many young minds get too engaged and even addicted. The risk of getting addicted to reading school books is much smaller.
It is natural to ask if pedagogics can learn something from computer games? Yes, I believe so. In particular, the Body&Soul reformed mathematics education combining analytical and computational mathematics discussed in previous blogs, can be structured like a computer game including the essential aspects of successively acquiring skills to meet new challenges, with a lot of feed-back. Since computer game technology largely is computational mathematics, it can be fruitful to teach computational mathematics as a form of computer game, or more generally mathematics as a form of IT.
A goal of Icarus Simulation is to develop an interactive web-based version of the Body&Soul program, with features of computer games, to be used in the new Bachelors program in Simulation Technology at KTH.
If we view interactive simulation as a form of computer game, then we can describe Simulation Technology as an education in the design and construction of computer games based on realistic simulation of physical phenomena, compactly described as
- an interactive computer game about how to construct interactive computer games
which includes mathematics, computation, programming, visualization, physics, mechanics...
More generally, theoretical science can be seen as a game against Nature with the goal of revealing, describing and simulating the secrets of Nature using the language of mathematics...
an addicitive for scientists...
torsdagen den 10:e september 2009
Incremental or Conservation Party?
Continuing the discussion from the previous post, let us note that the Navier-Stokes equation
expressing conservation of momentum, alternatively can be expressed as Newton's 2nd law
F = ma = m dv/dt
with F the force acting on an element of fluid of mass m and acceleration a = dv/dt. We can view these formulations to be equivalent from analytical mathematical point of view, but we may ask if they also are equivalent physically, or computationally?
Of course they are equivalent, you may say, because mathematics rules the game, but it is not so simple and clear if we recall that the Navier-Stokes equations cannot be solved exactly analytically, only approximately digitally by computers. The equivalence is then not so clear anymore.
So which formulation is most suitable to computation? Newton's 2nd law because it can be solved by time-stepping moving forward in time with small increments of time: The force F gives the acceleration dv/dt = F/m which tells the change of velocity which tells the change in position, from one time level to the next.
On the other hand, conservation of momentum is not directly ready for time-stepping, since it just expresses that something is conserved, namely momentum.
We are thus led to prefer an interpretation of a law of nature, which is most accessible to computation. We may prefer such an interpretation also from physical point of view, if we view real physics as some form of analog computation, as discussed in the knol Is the World a Computation?
Light refraction is a result of the wavelike nature of light as propagating electromagnetic waves. Light refraction can alternatively be described as shortest time of travel of light rays. Wave propagation can be time-stepped, while shortest time of travel is a global minimization problem, for which computational solution is less direct. We are led to view light as waves from physical and computational point of view, rather than as rays of particles.
An equilibrium states may be described as a state of balance of forces without any net force driving change. To find an equilibrium state of a system, we may time-step the system starting from some out-of-balance non-equilibrium state, with the hope that the system by itself approaches equilibrium. A physical law could then express the dynamics of a system computable by time-stepping, rather than a balance of forces at equilibrium, since this balance may not be directly computable.
A minimization principle in physics, like minimal time of travel of light, would then not qualify as a physical law unless augmented by e.g. time-stepping into computable form.
Computational solution of the Navier-Stokes equations.
In his 1963 Nobel Lecture discussed in the previous post: Events, Laws of Nature and Invariance Principles, Eugene Wigner expresses the physicist dream of a Theory of Everything TOE as some fundamental invariance principle or conservation law in the form of a differential equation, to which the World would be the solution.
If we narrow down the World to fluid mechanics, which is a reasonable a approximation as concerns macroscopic phenomena, then we already have a TOE of fluid mechanics
in the form of the Navier-Stokes equations expressing conservation of mass, momentum and energy. From Wigner's point of view this would close the scientific field of fluid mechanics since everything there is to know, is known: The Navier-Stokes equations!
This TOE would seem to represent extremely effective knowledge, since the NS equations can written down in two lines and can be taught to most people in less than an hour. It would be like a very compact two-line genetic code of fluid mechanics.
But thus is too simple, you say, right? Fluid mechanics is more than just jotting down the NS equations! Yes, you are right! The NS equations also have to be solved to tell us anything, and that turns out to be impossible by analytical mathematics: Only very simple analytical solutions are known which tell you very little about fluid mechanics. So the NS equations alone is not a TOE.
But computing solutions of the NS eqautions numerically using computers is today possible, because computers are now powerful enough, which in a sense gives you a TOE for fluid mechanics. However, there is hook: You have to compute solutions one by one with different data and you cannot get all solutions in one shot.
Today you have a wonderful laboratory in your laptop allowing you to explore the rich field of fluid mechanics successively by computing solutions of the NS equatiopns, studying their properties and hopefully discovering regularities or even laws supporting understanding. Some of what you can learn from this laboratory is presented on my knols on Fluid Mechanics.
Wigner's vision of a TOE represents a pre-computer classical approach to physics, which is beyond reach because even if the basic equation is a simple analytical equation like NS, the World as the solution to the equation is not simple and cannot be described by analytical mathematics, at least not a priori. This does not say that an a posteriori description by analytical mathematics is also impossible. Once solutions have been computed one can start to look for regularities and maybe find some which can be expressed by analytical mathematics, but only a posteriori. Wigner without computer is unreasonable.
In the previous post A Critical Analysis of the Ideology of Mathematics we made the observation that the foundation of school mathematics on all levels can be expressed in the words of Physics Nobel Laurate Eugene Wigner as:
- The unreasonable effectiveness of mathematics in the natural sciences. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
In his 1963 Nobel Lecture Events, Laws of Nature and Invariance Principles Wigner further explains:
- Physics does not endeavor to explain nature, it only endeavors to explain the regularities in the behavior of objects, which are called the laws of nature.
Acoording to Wigner mathematics is thus unreasonably effective as a language of expressing laws of nature interpreted as regularities of nature.
But is this unreasonable? Is it not completely reasonable that analytical mathematics is effective in expressing regularities, like an elliptic orbit or harmonic oscillation? So if laws of nature express regularities it is fully reasonable that they can be expressed in the language of analytical mathematics.
But what is a law of nature? Is it really a regularity expressible by analytical mathematics as Wigner seems to claim?
Let us take the same example as Wigner: Consider at planetary system governed by Newton's laws of motion, which no doubt are laws of nature. Is this all there is to say about planetary systems? No, it is not because the planetary motions are not included in Newton's laws.
The motions result from letting the system evolve forward in time step by step according to Newton's laws from some initial state. In the simplest case of one planet orbiting a heavy sun the orbit is an ellipse, but with more than one planet the motion can be abitrarily complicated and not allow a representation in terms of elementary functions.
Does this mean that there are no laws for the motion of a planetary system with many planets?
Of course not, but these laws are not explicit to us like Newton's laws, but hidden implicit and there is no golden rule how to find them and express them by elementary fucntions.
We may compare with turbulent flow which is governed by Newton's laws but exhibits a very complex partly chaotic structure with a variety of interacting vortices on different scales. But even a turbulent flow can exhibit some regularities in the form of certain meanvalues, which can be computationally predicted even if pointvalues vary chaotically, meanvalues like drag and lift. However, there is no neat mathematical formula that expresses the drag and lift of a given body. Turbulent flow has to be computed step by step and there is no shortcut to regularity of solutions as in the case of the elliptic orbit of one planet around a sun.
We are led to the conclusion analytical mathematics is not unresonably effective but rather resonably ineffective, while computational mathematics is resonably effective.
onsdagen den 9:e september 2009
Crash simulation of school mathematics.
In the Fall 2010 a new Bachelors Program in Simulation Technology will start at the Royal Institute of Technology KTH based on the Body&Soul Applied Mathematics Reform Program.
In the new B&S program standard analytical mathematics of calculus and linear algebra is combined with computational calculus and linear algebra into an integrated synthesis of analytical and computational mathematics, which opens entirely new possibilities in teaching and learning in science, technology, simulation, visualization and virtual reality.
This is because calculus and linear algebra boosted by computers gives a new very powerful tool allowing simulation of complex phenomena of real and imagined worlds unreachable by analytical mathematics.
The shift from standard analytical mathematics to computational mathematics has met strong opposition from the mathematics department at KTH unable or unwilling to reform standard analytical mathematics courses. The fact that KTH anyway has decided to start a program based on computational mathematics, given by people outside the mathematics department, opens to a similar reform in engineering education as a whole...
Connecting to the previous blogpost Will Mathematics be Replaced by IT in School Education?
one can see the shift at KTH from analytical to computational mathematics as replacing standard analytical mathematics by IT, a shift which can propagate down through the whole school system....
The consequences of the KTH decision thus can be far-reaching, since KTH is a leading university and sets the agenda for school mathematics...in Sweden at least...
As noted in the previous blog, mathematics education of today is in many ways similar to the education in religion of yesterday, which is no longer mandatory in Western schools.
From this experience we may expect that mathematics will not be mandatory tomorrow. Does this mean that students will no longer learn any mathematics. Not at all!
A new subject is now entering education on all levels: Information Technology or IT. Most likely IT will replace mathematics as the core of education together with language. But IT is largely based on logic, programming and computational mathematics, and it is possible to envision an IT education which teaches more mathematics than the present system does. More relevant mathematics for everybody and much more for students with special interest in mathematics and IT.
Integrating mathematics education with IT education is the logical conclusion of the mathematical war in the 1930s, which was won by the constructivists when Gödel hit the logicist/formalist school with his incompleteness theorems. After 80 years of incubation with a rise of the IT age, Gödel's poison now starts to have an effect. After all, IT and constructivist/computational mathematics is the same.
tisdagen den 8:e september 2009
The Ideology of Mathematics as presented in the documents of the previous blogpost can be summarized as:
- Mathematics has a double character: It is both the most original, complex and beautiful free creation of the human spirit with its own internal standards, and a universal practical tool. The miraculous double character is described by the Physics Nobel Laurate Eugene Wigner as:
- The unreasonable effectiveness of mathematics in the natural sciences. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
Mathematics education is based on this mystification, this true miracle:
It allows mathematics education from early grades and on to focus on the language and logic of mathematics learning the words "and", "or", "not", "there is/are", "for some", "for every", "for all", in the safe conviction that this will be most useful to all students in their practical lives.
It allows mathematics education to focus on axiomatics in the safe conviction that human knowledge can be axiomatized.
It allows mathematics education to focus just about anything in the safe conviction that mathematics is universally applicable.
But there are no miracles, trivialities are trivialities, axiomatization of knowledge is impossible, and no mathematics is universally applicable. To build education on a mystery which is neither understood nor deserved, is not a good idea, because learning is about understanding and students do not profit from free gifts which they do not understand nor deserve and which they cannot use.
For a discussion of how mathematics education can and should be reformed see my blog posts
showing close similarities between mathematics and religion sharing the double character of uplifted divinity and universal practicality. In both cases the idea is to study the language of the Divinity to learn about the World. This is still practiced in Islamic schools, but no longer in Western schools.
The two characters of mathematics clashed in the great war between the logicist/formalist and constructivist schools in 1920-30s. The constructivists won on technical knockout but they were soon cleansed from mathematics departments filled with logicists/formalists still in control and forming the ideology of mathematics today, so well expressed by the Committe on Logical Education. For more war reports, see my knols on mathematics.
What is the ideology of mathematics underlying mathematics education as presented by professional mathematicians? Let us seek an answer in the following typical texts to be analyzed in the next blogpost:
- Mathematics relies on both logic and creativity, and it is pursued both for a variety of practical purposes and for its intrinsic interest.
- For some people, and not only professional mathematicians, the essence of mathematics lies in its beauty and its intellectual challenge.
- For others, including many scientists and engineers, the chief value of mathematics is how it applies to their own work.
- Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is requisite for scientific literacy. To achieve this, students need to perceive mathematics as part of the scientific endeavor, comprehend the nature of mathematical thinking, and become familiar with key mathematical ideas and skills.
The Committee on Logic Eduction offers the following comprehensive summary:
- There are creative tensions in mathematics between beauty and utility, abstraction and application, between a search for unity and a desire to treat phenomena comprehensively.
- Mathematics was originally linked with science and technology; however, it gradually became independent of science and technology, and present-day mathematicians think freely about virtually everything possible.
- Therefore, mathematics is said to be a free creation of the human spirit. Special characteristics of mathematics are the clarity and precision of definitions, including usuage of words in ways that differ from their use in everyday language, and the certainty of mathematical truth based on deductive mathematical reasoning.
- Given what Wigner call the "unreasonable effectiveness of mathematics", all students should learn the basic nature of mathematics and mathematical reasoning and its use in organizing and modeling natural phenomena.
- In the practice of mathematics, typically some concepts and statements are taken as given. They may be applied or serve as the foundation for the development of further mathematics. Additional concepts can be defined carefully in terms of the given ones. Conjectures can be developed on the basis of experience with examples. Further statements can be proved deductively based on what has been assumed.
- This process has been repeated extensively, resulting in mathematics having its own intricate structure, with concepts and areas of specialization that require considerable time and study to grasp. Moreover mathematics is interconnected in many interesting ways.
- It may be useful to think of students learning mathematics along the lines of a generalized structure of reasoning: (1) recognition, (2) analysis, (3) informal deduction, (4) formal deduction, (5) axiomatics.
- In early grades, students learn the basic language including the critical logical words "and", "or", "not", "there is/are", "for some", "for every", "for all". They see multidigit numbers being built from single digit numbers. They match the trajectory of a kicked ball with the concept of line. They recognize patterns in sequences of numbers and shapes. In middle grades students develop habits of reasoning "locally", clarifying the assumptions of a particular problems and examining the steps involved in the solution to determine correctness.
- For example, one of us recently observed a fifth grade teacher asking her students for the definition of polygon. They knew, for example that triangles, squares and hexagons were polygons. It was exciting to see the students wrestling with abstraction, differentiating polygons from circles, and finally focusing on polygons as figures with sides.
- By the end of high school, students should be aware of the global deductive nature of axiomatic mathematics. They should be familiar with the connections between our number systems and algebra, between algebra and geometry.
- They should be comfortable reasoning with short sequences of statements, with Venn diagrams and other visual and diagrammatic methods. They should have experience with modeling, recognizing for example that certain natural phenomena obey linear relationships and that linear relationships make prediction so easy that we try to approximate other more complicated phenomena by linear ones.
- It is important both to understand how algebraic relationships can describe particular problems and to understand the power derived by working abstractly with the mathematics which applies to many different situations.
Everybody Counts: A Report to the Nation on the Future of Mathematics Education describes The Nature of Mathematics as follows:
- Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems.
- As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.
- The special role of mathematics in education is a consequence of its universal applicability. The results of mathematics--theorems and theories--are both significant and useful; the best results are also elegant and deep. Through its theorems, mathematics offers science both a foundation of truth and a standard of certainty.
- In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power--a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives.
- Mathematics empowers us to understand better the information-laden world in which we live. During the first half of the twentieth century, mathematical growth was stimulated primarily by the power of abstraction and deduction, climaxing more than two centuries of effort to extract full benefit from the mathematical principles of physical science formulated by Isaac Newton.
- Now, as the century closes, the historic alliances of mathematics with science are expanding rapidly; the highly developed legacy of classical mathematical theory is being put to broad and often stunning use in a vast mathematical landscape.
- Several particular events triggered periods of explosive growth. The Second World War forced development of many new and powerful methods of applied mathematics. Postwar government investment in mathematics, fueled by Sputnik, accelerated growth in both education and research. Then the development of electronic computing moved mathematics toward an algorithmic perspective even as it provided mathematicians with a powerful tool for exploring patterns and testing conjectures.
- At the end of the nineteenth century, the axiomatization of mathematics on a foundation of logic and sets made possible grand theories of algebra, analysis, and topology whose synthesis dominated mathematics research and teaching for the first two thirds of the twentieth century.
- These traditional areas have now been supplemented by major developments in other mathematical sciences--in number theory, logic, statistics, operations research, probability, computation, geometry, and combinatorics. In each of these subdisciplines, applications parallel theory.
- Even the most esoteric and abstract parts of mathematics--number theory and logic, for example--are now used routinely in applications (for example, in computer science and cryptography). Fifty years ago, the leading British mathematician G.H. Hardy could boast that number theory was the most pure and least useful part of mathematics. Today, Hardy's mathematics is studied as an essential prerequisite to many applications, including control of automated systems, data transmission from remote satellites, protection of financial records, and efficient algorithms for computation.
- In 1960, at a time when theoretical physics was the central jewel in the crown of applied mathematics, Eugene Wigner wrote about the ``unreasonable effectiveness'' of mathematics in the natural sciences: ``The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.''
- Theoretical physics has continued to adopt (and occasionally invent) increasingly abstract mathematical models as the foundation for current theories. For example, Lie groups and gauge theories--exotic expressions of symmetry--are fundamental tools in the physicist's search for a unified theory of force.During this same period, however, striking applications of mathematics have emerged across the entire landscape of natural, behavioral, and social sciences.
- All advances in design, control, and efficiency of modern airliners depend on sophisticated mathematical models that simulate performance before prototypes are built. From medical technology (CAT scanners) to economic planning (input/output models of economic behavior), from genetics (decoding of DNA) to geology (locating oil reserves), mathematics has made an indelible imprint on every part of modern science, even as science itself has stimulated the growth of many branches of mathematics.
- Applications of one part of mathematics to another--of geometry to analysis, of probability to number theory--provide renewed evidence of the fundamental unity of mathematics. Despite frequent connections among problems in science and mathematics, the constant discovery of new alliances retains a surprising degree of unpredictability and serendipity.
- Whether planned or unplanned, the cross-fertilization between science and mathematics in problems, theories, and concepts has rarely been greater than it is now, in this last quarter of the twentieth century.
The mathematician A N Whitehead, who wrote the bible of the logicist school Principia Mathematica together with Bertrand Russell, explains to us in Mathematics in the History of Thought:
- The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit.
- The originality of mathematics consists in the fact that in mathematical science connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious.
- Suppose we project our imagination backwards through many thousands of years, and endeavour to realise the simple-mindedness of even the greatest intellects in those early societies. Abstract ideas which to us are immediately obvious must have been, for them, matters only of the most dim apprehension. For example take the question of number.
- We think of the number 'five' as applying to appropriate groups of any entities whatsoever-to five fishes, five children, five apples, five days. Thus in considering the relations of the number 'five' to the number 'three,' we are thinking of two groups of things, one with five members and the other with three members. But we are entirely abstracting from any consideration of any particular entities, or even of any particular sorts of entities, which go to make up the membership of either of the two groups. We are merely thinking of those relationships between those two groups which are entirely independent of the individual essences of any of the members of either group. This is a very remarkable feat of abstraction; and it must have taken ages for the human race to rise to it.
- During a long period, groups of fishes will have been compared to each other in respect to their multiplicity, and groups of days to each other. But the first man who noticed the analogy between a group of seven fishes and a group of seven days made a notable advance in the history of thought. He was the first man who entertained a concept belonging to the science of pure mathematics. At that moment it must have been impossible for him to divine the complexity and subtlety of these abstract mathematical ideas which were waiting for discovery. Nor could he have guessed that these notions would exert a widespread fascination in each succeeding generation.
- The tremendous future effect of mathematical knowledge on the lives of men, on their daily avocations, on their habitual thoughts, on the organization of society, must have been even more completely shrouded from the foresight of those early thinkers. Even now there is a very wavering grasp of the true position of mathematics as an element in the history of thought.
- When we think of mathematics, we have in our mind a science devoted to the exploration of number, quantity, geometry, and in modern times also including investigation into yet more abstract concepts of order, and into analogous types of purely logical relations. The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction.
- All you assert is, that reason insists on the admission that, if any entities whatever have any relations which satisfy such-and-such purely abstract conditions, then they must have other relations which satisfy other purely abstract conditions.
- Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about. So far is this view of mathematics from being obvious, that we can easily assure ourselves that it is not, even now, generally understood.