The 50 Greatest Breakthroughs Since the Wheel      

Why did it take so long to invent the wheelbarrow? Have we hit peak innovation? What our list reveals about imagination, optimism, and the nature of progress.

The Atlantic asked a dozen scientists, historians, and technologists to rank the top innovations since the wheel. Here are the results.

You can also choose your own top five innovations, and see how the readers' choices stack up against the Atlantic experts'.

James Fallows, 10/23/13

1.     The printing press, 1430s

The printing press was nominated by 10 of our 12 panelists, five of whom ranked it in their top three. Dyson described its invention as the turning point at which “knowledge began freely replicating and quickly assumed a life of its own.”

2.     Electricity, late 19th century

And then there was light -- and Nos. 4, 9, 16, 24, 28, 44, 45, and most of the rest of modern life.

3.     Penicillin, 1928

Accidentally discovered in 1928, though antibiotics were not widely distributed until after World War II, when they became the silver bullet for any number of formerly deadly diseases

4. Semiconductor electronics, mid-20th century

The physical foundation of the virtual world

5. Optical lenses, 13th century

Refracting light through glass is one of those simple ideas that took a mysteriously long time to catch on. “The Romans had a glass industry, and there’s even a passage in Seneca about the optical effects of a glass bowl of water,” says Mokyr. But it was centuries before the invention of eyeglasses dramatically raised the collective human IQ, and eventually led to the creation of the microscope and the telescope.

6. Paper, second century

“The idea of stamping images is natural if you have paper, but until then, it’s economically unaffordable.” -- Charles C. Mann

7. The internal combustion engine, late 19th century

Turned air and fuel into power, eventually replacing the steam engine (No. 10)

8. Vaccination, 1796

The British doctor Edward Jenner used the cowpox virus to protect against smallpox in 1796, but it wasn’t until Louis Pasteur developed a rabies vaccine in 1885 that medicine -- and government -- began to accept the idea that making someone sick could prevent further sickness.

9. The Internet, 1960s

The infrastructure of the digital age

10. The steam engine, 1712

Powered the factories, trains, and ships that drove the Industrial Revolution

11. Nitrogen fixation, 1918

The German chemist Fritz Haber, also the father of chemical weapons, won a Nobel Prize for his development of the ammonia-synthesis process, which was used to create a new class of fertilizers central to the green revolution (No. 22).

12. Sanitation systems, mid-19th century

A major reason we live 40 years longer than we did in 1880 (see “Die Another Day”)

13. Refrigeration, 1850s

“Discovering how to make cold would change the way we eat -- and live -- almost as profoundly as discovering how to cook.” -- George Dyson

14. Gunpowder, 10th century

Outsourced killing to a machine

15. The airplane, 1903

Transformed travel, warfare, and our view of the world (see No. 40)

16. The personal computer, 1970s

Like the lever (No. 48) and the abacus (No. 43), it augmented human capabilities.

17. The compass, 12th century

Oriented us, even at sea

18. The automobile, late 19th century

Transformed daily life, our culture, and our landscape

19. Industrial steelmaking, 1850s

Mass-produced steel, made possible by a method known as the Bessemer process, became the basis of modern industry.

20. The pill, 1960

Launched a social revolution

21. Nuclear fission, 1939

Gave humans new power for destruction, and creation

22. The green revolution, mid-20th century

Combining technologies like synthetic fertilizers (No. 11) and scientific plant breeding (No. 38) hugely increased the world’s food output. Norman Borlaug, the agricultural economist who devised this approach, has been credited with saving more than 1 billion people from starvation.

23. The sextant, 1757

It made maps out of stars.

24. The telephone, 1876

Allowed our voices to travel

25. Alphabetization, first millennium b.c.

Made knowledge accessible and searchable -- and may have contributed to the rise of societies that used phonetic letters over those that used ideographic ones

26. The telegraph, 1837

Before it, Joel Mokyr says, “information could move no faster than a man on horseback.”

27. The mechanized clock, 15th century

It quantified time.

28. Radio, 1906

The first demonstration of electronic mass media’s power to spread ideas and homogenize culture

29. Photography, early 19th century

Changed journalism, art, culture, and how we see ourselves

30. The moldboard plow, 18th century

The first plow that not only dug soil up but turned it over, allowing for the cultivation of harder ground. Without it, agriculture as we know it would not exist in northern Europe or the American Midwest.

31. Archimedes’ screw, third century b.c.

The Greek scientist is believed to have designed one of the first water pumps, a rotating corkscrew that pushed water up a tube. It transformed irrigation and remains in use today at many sewage-treatment plants.

32. The cotton gin, 1793

Institutionalized the cotton industry -- and slavery -- in the American South

33. Pasteurization, 1863

One of the first practical applications of Louis Pasteur’s germ theory, this method for using heat to sterilize wine, beer, and milk is widely considered to be one of history’s most effective public-health interventions.

34. The Gregorian calendar, 1582

Debugged the Julian calendar, jumping ahead 10 days to synchronize the world with the seasons

35. Oil refining, mid-19th century

Without it, oil drilling (No. 39) would be pointless.

36. The steam turbine, 1884

A less heralded cousin of steam engines (No. 10), turbines are the backbone of today’s energy infrastructure: they generate 80 percent of the world’s power.

37. Cement, first millennium b.c.

The foundation of civilization. Literally.

38. Scientific plant breeding, 1920s

Humans have been manipulating plant species for nearly as long as we’ve grown them, but it wasn’t until early-20th-century scientists discovered a forgotten 1866 paper by the Austrian botanist Gregor Mendel that we figured out how plant breeding -- and, later on, human genetics -- worked.

39. Oil drilling, 1859

Fueled the modern economy, established its geopolitics, and changed the climate

40. The sailboat, fourth millennium b.c.

Transformed travel, warfare, and our view of the world (see No. 15)

41. Rocketry, 1926

“Our only way off the planet -- so far.” -- George Dyson

42. Paper money, 11th century

The abstraction at the core of the modern economy

43. The abacus, third millennium b.c.

One of the first devices to augment human intelligence

44. Air-conditioning, 1902

Would you start a business in Houston or Bangalore without it?

45. Television, early 20th century

Brought the world into people’s homes

46. Anesthesia, 1846

In response to the first public demonstration of ether, Oliver Wendell Holmes Sr. wrote: “The fierce extremity of suffering has been steeped in the waters of forgetfulness, and the deepest furrow in the knotted brow of agony has been smoothed for ever.”

47. The nail, second millennium b.c.

“Extended lives by enabling people to have shelter.” -- Leslie Berlin

48. The lever, third millennium b.c.

The Egyptians had not yet discovered the wheel when they built their pyramids; they are thought to have relied heavily on levers.

49. The assembly line, 1913

Turned a craft-based economy into a mass-market one

50. The combine harvester, 1930s

Mechanized the farm, freeing people to do new types of work

Of CRISPR/Cas and the power of basic research             

Researchers didn’t set out to discover a genome-editing technique. But they found one anyway.  

Jeffrey Perkel, 09/18/13

It’s not often one literally gets to see history made in the scientific arena. But that’s what happened in the summer of 2012, in the pages of Science magazine.

Jennifer Doudna of the University of California, Berkeley, and Emmanuelle Charpentier, then at Umeå University in Sweden, demonstrated the molecular mechanics underlying a quirky bacterial system called CRISPR/Cas. More to the point, they demonstrated that the system could be reprogrammed to do something incredibly useful, genetically speaking: a process called genome editing.

That such a system exists naturally and can be harnessed for the good of the research community, seems almost incredible. As Harvard University geneticist George Church, who uses the system, told me in August, “It’s a real gift from biology.” But it also is proof of the power of basic biological research, as the power of CRISPR/Cas is something nobody could have predicted at the outset.

In brief – and I’ll expand on this below – the CRISPR/Cas system gives researchers a tool to make specific, surgical changes in the genome of living cells.

The CRISPR/Cas system. (From E. Pennisi, Science, 341:833-6, 2013. Credit: K. Sutliff/Science) (請至原網頁參考解說圖片)

Suppose, for instance, that researchers wanted to repair the genetic defect in an individual with Huntington’s disease. Using the Nobel Prize-winning development known as “induced pluripotent stem (iPS) cells,” they could take cells from this individual and convert them into a kind of embryonic stem cell, which can develop into any kind of cell or tissue in the body. They could then use the CRISPR/Cas system to repair the defect in those cells, convert those cells into neurons and, in theory anyway, transplant them back into the patient.

Now, that’s all a LONG WAY OFF. But that’s the promise of CRISPR/Cas, and it wasn’t lost on the scientific community.

Within months of Doudna and Charpentier’s original paper, the scientific literature was awash in papers extending their findings. Their paper has been cited 78 times, according to the Web of Science, a database that tracks such things – that’s a huge number for just one year. (By comparison, the average paper is cited perhaps once in a year, and even in the high-powered journal Science, a typical paper is cited about 30 times in two years.)

In recognition of the frenzy the discovery has set in motion, Science in August published a nice overview of the research to date, entitled “The CRISPR Craze.”

So, just what is CRISPR/Cas? Well, first of all, it’s a clumsy acronym. CRISPR is short for “Clustered Regularly Interspaced Short Palindromic Repeats;” Cas is “CRISPR-associated.”

Basically, as Church explained it to me, CRISPR/Cas represents a kind of bacterial adaptive immune system, like your B- and T-cells.

At the heart of the system is a series of repeating sequences in the genomes of bacteria. Researchers had for years noticed these funny elements, but had no idea what they might be doing. Then one day, somebody realized they were similar to the genetic material of viruses that infected bacteria. That led to the recognition that when bacteria were infected with a pathogen, they recorded the event in their genetic code, enabling them to better recognize and fight the pathogen in the future.

How do they fight those pathogens? Enter Cas. This is an enzyme that makes cuts in DNA, but not just anywhere. Cas cuts DNA at a sequence specified by the CRISPR elements. The CRISPR sequences are copied into RNA, which the cell then dices up into short pieces; these seek out their matching sequences in the pathogen’s genome, and Cas goes to work, inactivating the invading genetic material.

The DNA cut produced by Cas can be repaired by either of two mechanisms. The simpler one, called non-homologous end-joining, simply tries to suture the two cut ends together again, an error-prone process. The second method, homology-directed repair, uses a backup copy of the gene as a template to fix the damaged copy. Researchers can supply their own template sequences to direct this process to, for instance, insert a gene that would cause the cells to fluoresce whenever a specific gene is turned on.

Now, to be clear there are other systems that do the same thing, particularly zinc finger nucleases (ZFNs) and TALE nucleases (TALENs). These are man-made proteins that can be programmed to target specific DNA elements, no guide RNA required. They have proven to be exceptionally useful research tools, and some are even being tested in clinical trials. But they’re also expensive, tedious, and challenging to build, and not every ZFN or TALEN cuts as expected.

CRISPR/Cas essentially has none of those limitations – anyone can order the protein and a short targeting RNA sequence (a relatively inexpensive proposition), and by all accounts, the process just seems to work.

In their 2012 paper, Doudna and Charpentier’s team demonstrated that Cas uses two short RNAs, called a crRNA and tracrRNA, to work its magic. Those two RNAs could be linked into a single molecule, called a guide RNA, making the system a simple two-component tool.

Then – and this is the important point – they demonstrated they could direct Cas to a sequence of their choosing by adding an appropriate guide molecule.

They did that in a test tube, but it wasn’t long before others demonstrated the process also works in live cells. In January, six papers were published (including one from Doudna’s lab) showing that CRISPR/Cas can make genetic alterations in human cells, mouse cells, bacterial cells, and even in zebrafish.

Since then, other researchers have replicated and extended these findings. Some have shown, for instance, that the system is “multiplexable” – that is, users can make multiple genetic modifications in the same cell by supplying more than one guide RNA, something that isn’t possible with TALENs and ZFNs. (One paper demonstrated it was possible to make five alterations simultaneously.) Others have demonstrated they can use the CRISPR/Cas system, with some judicious protein modifications, to alter the expression of genes, using the system to target genetic regulatory proteins to specific genomic addresses. (That latter point is graphically illustrated at the bottom of the figure above.)

A number of researchers have tackled the question of CRISPR/Cas’s specificity – that is, how likely is it to modify an unintended sequence along with the user’s desired target. (Short answer: The system does have some issues with off-target effects, but they can be addressed using relatively simple tweaks and workarounds; one approach is described here.) Some are exploring the therapeutic applications of the technology. And at least two papers (here and here) have shown that it is possible to make genetic alterations in the iPS cells I mentioned earlier, a precursor to the idea of genetic repair.

It’s a remarkable string of developments in so short a time. And as Doudna said when I spoke with her in March, it arose not from a goal-oriented research project — somebody saying, hey guys, let’s go invent a useful genetic tool! — but basic research, a desire to understand something new and interesting:

“I think it’s a really great example of how fundamental basic research, in this case, which was not aimed at any particular target or goal or certainly a particular application, led to the discovery of a system that may turn out to be a really transformative technology for genome engineering.”

Given the staggering difficulties researchers face getting funding for basic research projects, somebody in Washington would do well to remember that.

Jeffrey Perkel is the DXS tech editor and a recovering scientist who has always had a passion for the technology and the gadgetry of science. He has been a scientific writer and editor since 2000, when he left academia to join the staff of The Scientist magazine as a Senior Editor for Technology. Before that, he studied transcription factor biology at the University of Pennsylvania and Harvard Medical School -- training that, surprisingly, has little application in the real world. In 2006, he and his family headed west to Pocatello, Idaho, and has been a freelance writer ever since. You can see why Double X Science is thrilled to have him on the team!

Comment 1

AnonAuth

What a poorly constructed list! What about:

- Regenative medicine, 3D printed body parts and tissue scaffolding.
- Nanotechnology, graphene, CNT and other novel electronic and photonic materials.
- Quantum computing, cryptography and information processing.
- Microfluidics and lab-on-chip technologies.
- Spintronics

Comment 2

artur

How can you leave out neuroscience?

Comment 3

HenryC

What is the source of the hottest fields of research, because it is no where near accurate. In number of papers, and scientists involved nano technology dwarfs the others. Every one of the subjects are very interesting, but nowhere near the hottest.

http://www.wired.com/wiredscience/2013/08/the-10-hottest-fields-of-science-research/?pid=10841&viewall=true

The 10 Hottest Fields of Science Research               

Jeffrey Marlow, 08/20/13

Scientific research is a large and sprawling endeavor, with thousands of laboratories around the world studying their own ultra-specialized piece of a much more significant whole. It’s the logical intersection of reductionist scientific heritage and centuries of technological advances: in order to advance our understanding of the world around us, we must pursue increasingly specific sub-disciplines, from retina neural computation to space plasma physics.

Which is why Thomson Reuters’ scene-scoping study on “100 Key Scientific Research Fronts” is a welcome report for science enthusiasts eager to stay updated on cutting-edge research but lacking the time to read every issue of Science or Nature cover-to-cover.

The report ranks research areas with a special sauce formula that first divides the entirety of scientific research into 8,000 categories that form the “Thomson Reuters Essential Science Indicators” database. Within each subdivision, a set of core papers is designated by frequent and clustered citations, identifying foundational scientific literature that earned a lot of shout outs in reports of subsequent discoveries. To find today’s hottest research fields, only core papers published between 2007-2012 were considered; the number of citations of those papers and their average publication date were compiled. As the report notes, “a research front with many core papers of recent vintage often indicates a fast-moving or hot specialty.”

This doesn’t necessarily mean these fields are the most important or the most beneficial to society – it just means scientists (and, by extension, groups funding the research) are getting pretty excited about what they’re learning. Here, we take a quick look at the hottest research front in each of ten thematic categories – the sharpest of the cutting edge.

Impact of Climate Change on Food Crops

Climate change is happening, and while current disruptions may mostly involve higher air conditioning bills, the very basis of the planet’s food supply may ultimately be at risk. Plant scientists are hard at work developing drought-resistant crop strains and determining how biomes will shift.

Tectonic Evolution of the Southern Central Asian Orogenic Belt

The Central Asian Orogenic Belt – a mountainous 5.3 million square kilometer region that includes this region of Uzbekistan – is a twisted amalgamation of re-purposed continental crust, sedimentary rocks, scraped-off seafloor sediments, and igneous rock. Just how it formed, however, is a matter of continued debate, with one team proposing continental collisions and another championing accretion of seafloor sediments during plate subductions.

Transcatheter Aortic Valve Implantation

In 2002, a French doctor replaced a patient’s heart valve not by surgically opening the chest cavity, but by catheter, requiring only a small incision. This new method has gained popularity in treating cases of faulty aortic valves, and has been approved for use in dozens of countries, including the United States.

DNA Methylation Analysis and Missing Heritability

DNA, the dogma goes, is the code of life, determining everything from hair color to our susceptibility to diseases. But many traits passed down from generation to generation are untraceable in the genome – a source of “missing heritability”. What if our environment changed gene expression, not only in our lifetime, but also in our children’s? Some studies are suggesting that this is precisely the case, potentially due to the addition of a CH3 group (methylation) to certain genes. It’s a controversial field, but one that will continue to examine the most essential questions of what our genetic code really means.

Ocean Acidification and Marine Ecosystems

Increased quantities of carbon dioxide in the atmosphere are generating higher concentrations of carbonic acid in the oceans, lowering the pH and dissolving some organisms’ shells or skeletons. How exactly the enormously complicated oceanic ecosystem – which remains poorly characterized – will respond to the shifting chemical milieu remains an open question. Above, a coral reef in Palmyra Atoll National Wildlife Refuge.

Enhanced Visible Light Photocatalytic Hydrogen Production

Hydrogen fuel cells have been “the next big thing” in energy for years, with proponents trumpeting their potential to drastically reduce our dependence on fossil fuels. But producing the hydrogen has been a stumbling block. New materials – complex catalysts often involving metals such as cobalt, nickel, iron, or molybdenum – have helped scientists learn more about the molecular mechanisms of water-splitting hydrogen production. With continued progress, futuristic fantasy of sunlight-powered cars may be resuscitated.

Alkali Doped Iron Selenide Superconductors

Superconductivity allows electrical current transmission with no resistance or loss of signal. Traditionally, this near-magical state could only be attained at extremely low temperatures (as in the photo above, using liquid nitrogen), permitting only limited applications, but researchers have been coaxing the temperature of superconductivity steadily upward. Over the last several years, iron-arsenic compounds have been replaced by iron-selenium devices, generating a lot of buzz in the physics community.

Galileon Cosmology

The universe is getting bigger, and it’s doing so at ever faster rates. One explanation is “dark energy,” but skeptics consider its invocation mildly unsatisfying; an alternative approach, they propose, is to modify our mathematical treatment of gravity’s effects at large distances. The use of a particular method – the galileon scalar field – allows for self-accelerating solutions, which may ultimately be able to reconcile one of the universe’s biggest mysteries.

High Energy Rechargeable Lithium Air Batteries

Lithium air batteries strip electrons from lithium and shuttle them to oxygen, using the resulting current to drive electrical devices. The use of air-based oxygen as an electron acceptor means you don’t need to store an oxidizer in the battery, allowing for high energy densities comparable to those of gasoline-powered engines. Above, IBM’s effort to produce a scalable lithium-air battery.

Urban Policy Mobilities and Global Governance Issues

Every week, more than one million people move from rural regions to cities. This massive shift – a relocation unlike anything in human history – represents a complete reconfiguration of how people interact with the planet. How governments respond, though security, infrastructure, or economic instruments, may well determine if the global urban future will more closely resemble dystopian shanty towns or gleaming metropolises that foster collaboration and increase societal efficiency. Above, a sea of skyscrapers in Shanghai.

(請至原網頁瀏覽參考圖片)

http://www.wired.com/wiredscience/2013/08/the-10-hottest-fields-of-science-research/?pid=10841&viewall=true

10 Greatest Ideas in the History of Science  

Scientific Revolutions

(接開欄文)

Symmetry Quantifies Beauty

Symmetry, that somewhat vague concept that involves folding or twisting triangles, cubes and other objects in various ways has applications far beyond high school geometry class. As it turns out, the universe is riddled with symmetry, or the lack thereof.

The most beautiful human faces are also the most symmetrical. Atoms in a crystal are arranged in a symmetrical, repeating pattern. Many other phenomena throughout nature exhibit breathtaking symmetry, from honeycombs to spiral galaxies.

Particle physics and astrophysics are also captivated by the concept of symmetry. One of the biggest asymmetries is the fact that our universe is made of more matter than antimatter. If the universe were perfectly symmetrical, there would be equal amounts of both. (But then the universe probably wouldn't exist since matter and antimatter annihilate each other.) However, as Atkins writes, the universe is symmetrical "if simultaneously we change particles for antiparticles..., reflect the universe in a mirror..., and reverse the direction of time."

Does that explain why Miss Universe is always so pretty?

Source: Galileo's Finger: The Ten Great Ideas of Science by Peter Atkins

Classical Mechanics Fails to Describe Small Particles

The classical physics of Isaac Newton and James Clerk Maxwell work reasonably well for most everyday applications. But classical physics is limited in the sense that it does not quite accurately depict reality.

The first inkling that something was seriously wrong came from analysis of blackbody radiation. Imagine a hot stove: It first starts out red, then turns white as it gets hotter. Classical physics was incapable of explaining this. Max Planck, however, had an idea: Perhaps the released energy came in little packets called "quanta." Instead of energy taking on continuous values, it instead takes on only discrete values. (Think of the difference between a ramp and a staircase; a person standing on a ramp can take on any height, while a person standing on a staircase only has certain discrete heights from which to choose.) As it turns out, these "quanta" of light energy are today known as photons. Thus, it was demonstrated that light, which until that time generally had been thought of as a wave, could also act like discrete particles.

Then along came Louis de Broglie who extended the concept:

All particles can act like waves, and all waves can act like particles.

Slam-dunk evidence for this idea came by way of the famous double-slit experiment, which conclusively showed that photons, electrons and even molecules like buckyballs exhibit wave-particle duality. (A lab confirmed the results of this experiment yet again in May 2013.)

These two concepts, quantization and wave-particle duality, form the core of the discipline known as quantum mechanics. Two other core concepts include the uncertainty principle (i.e., the inability to know various pairs of characteristics of a system with precision) and the wavefunction (which, when squared, gives the probability of finding a particle in a particular location). And what does all that give us? Schrödinger's cat, which is simultaneously dead and alive.

No wonder Stephen Hawking always reaches for his gun.

Source: Galileo's Finger: The Ten Great Ideas of Science by Peter Atkins

The Universe Is Expanding

About 13.8 billion years ago, the universe underwent a period of rapid expansion, known as cosmic inflation. Immediately after that was the Big Bang. (Yes, cosmic inflation occurred before the Big Bang; see this detailed explanation by our favorite astrophysicist, Ethan Siegel.) Ever since then, the universe has kept right on expanding.

We know the Big Bang occurred because of the telltale evidence it left behind: The cosmic microwave background radiation (CMB). As the universe expanded, the initial burst of light from the Big Bang got stretched. (Remember, light can be both a wave and a particle.) When light is stretched, the wavelength increases. Today, that light is no longer visible with the naked eye because it now inhabits the microwave range of the electromagnetic spectrum. However, you can still "see" it on old-school television sets with antennas; the static on "in-between" channels is partially due to the CMB.

But not only is the universe expanding, its rate of expansion appears to be accelerating due to dark energy. And the further away an object is from Earth, the faster it is accelerating away from us.

If you thought the universe was a lonely place now, just wait 100 billion years. Thanks to dark energy, we won't be able to see any stars beyond our own galaxy (which, at that time, will be a merger between the Milky Way and Andromeda galaxies).

Source: Galileo's Finger: The Ten Great Ideas of Science by Peter Atkins

Spacetime Is Curved by Matter

The fabric of our universe is spacetime, which consists of the three spatial dimensions (length, width and height) combined with the dimension of time. Imagine this fabric as a stretchy, rubber sheet. And then imagine placing a giant bowling ball on that sheet. The sheet would warp around the bowling ball, and any object placed near the bowling ball would roll toward it. This metaphor for Albert Einstein's theory of general relativity explains how gravity works. (Despite being Einstein's greatest achievement, general relativity is not for what he won the Nobel Prize; instead, the prize was awarded for his work on the photoelectric effect.)

But this wasn't Einstein's only contribution. He also came up with special relativity which describes how time slows down for moving objects, especially as they travel closer to the speed of light.

Interestingly, the effects of both general and special relativity must be taken into account for GPS satellites to work properly. If these effects were not considered, then the clocks on Earth and on the satellites would be out of sync, and consequently, the distances reported by the GPS unit would be wildly inaccurate. So, every time you use your smartphone to succesfully find the local Starbucks, give thanks to Albert Einstein.

Source: Galileo's Finger: The Ten Great Ideas of Science by Peter Atkins

Mathematics Is the Limit of Reason

Fundamentally, mathematics makes no sense. That probably doesn't come as a surprise to those of us who struggled in algebra or calculus. Though it is the language of science, the truth is that mathematics is built upon a cracked foundation.

For instance, consider a number. You think you know one when you see one, but it's rather difficult to define. (In that sense, numbers are like obscenity or pornography.) Not that mathematicians haven't tried to define numbers. The field of set theory is largely dedicated to such an endeavor, but it isn't without controversy.

Or consider infinity. Georg Cantor did and went crazy in the process. Counterintuitively, there is such a thing as one infinity being larger than another infinity. The rational numbers (those that can be expressed as a fraction) constitute one infinity, but irrational numbers (those that cannot be expressed as a fraction) constitute a larger infinity. A special type of irrational number, called the transcendental number, is particularly to blame for this. The most famous transcendental is pi, which can neither be expressed as a fraction nor as the solution to an algebraic equation. The digits which make up pi (3.14159265...) go on and on infinitely in no particular pattern. Most numbers are transcendental, like pi. And that yields a very bizarre conclusion: The natural numbers (1, 2, 3...) are incredibly rare. It's amazing that we can do any math whatsoever.

At its core, mathematics is intimately tied to philosophy. The most hotly debated questions, such as the existence and qualities of infinity, seem far more philosophical in nature than scientific. And thanks to Kurt Gödel, we know that an infinite number of mathematical expressions are probably true, but unprovable.

Such difficulties explain why, from an epistemological viewpoint, mathematics is so disturbing: It places a finite boundary on human reason.

Source: Galileo's Finger: The Ten Great Ideas of Science by Peter Atkins