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Is mathematics truly a language?

Is mathematics truly a language?

Mathematics is called the language of science. Italian astronomer and physicist Galileo Galilei is attributed with the quote, “Mathematics is the language in which God has written the universe.” Most likely this quote is a summary of his statement in Opere Il Saggiatore:

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

Yet, is mathematics truly a language, like English or Chinese? To answer the question, it helps to know what language is and how the vocabulary and grammar of mathematics is used to construct sentences.

What Is a Language?

There are multiple definitions of “language.” A language may be a system of words or codes used within a discipline. Language may refer to a system of communication using symbols or sounds. Linguist Noam Chomsky defines language as a set of sentences constructed using a finite set of elements. Some linguists believe language should be able to represent events and abstract concepts.

Whichever definition is used, a language contains the following components:

  • There must be a vocabulary of words or symbols.
  • Meaning must be attached to the words or symbols.
  • A language employs grammar, which is a set of rules that outline how vocabulary is used.
  • A syntax organizes symbols into linear structures or propositions.
  • A narrative or discourse consists of strings of syntatic propositions.
  • There must be (or have been) a group of people who use and understand the symbols.

Mathematics meets all of these requirements. The symbols, their meanings, syntax, and grammar are the same throughout the world. Mathematicians, scientists, and others use math to communicate concepts. Mathematics describes itself (a field called metamathematics), real-world phenomena, and abstract concepts.

Vocabulary and Syntax in Mathematics

The vocabulary of math draws from many different alphabets and includes symbols unique to math. A mathematical equation may be stated in words to form a sentence that has a noun and verb, just like a sentence in a spoken language. For example:

3 + 5 = 8

could be stated as, “Three added to five equals eight.”

Breaking this down, nouns in math include:

  • Arabic numerals (0, 5, 123.7)
  • Fractions (1⁄4, 5⁄9, 2 1⁄3)
  • Variables (a, b, c, x, y, z)
  • Expressions (3x, x2, 4 + x)
  • Diagrams or visual elements (circle, angle, triangle, tensor, matrix)
  • Infinity (∞)
  • Pi (π)
  • Imaginary numbers (i, -i)
  • The speed of light (c)

Verbs include symbols including:

  • Equalities or inequalities (=, <, >)
  • Actions such as addition, subtraction, multiplication, and division (+, -, x or *, ÷ or /)
  • Other operations (sin, cos, tan, sec)

If you try to perform a sentence diagram on a mathematical sentence, you’ll find infinitives, conjunctions, adjectives, etc. As in other languages, the role played by a symbol depends on its context.

Mathematics grammar and syntax, like vocabulary, are international. No matter what country you’re from or what language you speak, the structure of the mathematical language is the same.

  • Formulas are read from left to right.
  • The Latin alphabet is used for parameters and variables. To some extent, the Greek alphabet is also used. Integers are usually drawn from i, j, k, l, m, n. Real numbers are represented by abc, α, β, γ. Complex numbers are indicated by wand z. Unknowns are x, y, z. Names of functions are usually f, g, h.
  • The Greek alphabet is used to represent specific concepts. For example, λ is used to indicate wavelength and ρ means density.
  • Parentheses and brackets indicate the order in which the symbols interact.
  • The way functions, integrals, and derivatives are phrased is uniform.

Language as a Teaching Tool

Understanding how mathematical sentences work is helpful when teaching or learning math. Students often find numbers and symbols intimidating, so putting an equation into a familiar language makes the subject more approachable. Basically, it’s like translating a foreign language into a known one.

While students typically dislike word problems, extracting the nouns, verbs, and modifiers from a spoken/written language and translating them into a mathematical equation is a valuable skill to have. Word problems improve comprehension and increase problem-solving skills.

Because mathematics is the same all over the world, math can act as a universal language. A phrase or formula has the same meaning, regardless of other language that accompanies it. In this way, math helps people learn and communicate, even if other communication barriers exist.

The Argument Against Math as a Language

Not everyone agrees that mathematics is a language. Some definitions of “language” describe it as a spoken form of communication. Mathematics is a written form of communication. While it may be easy to read a simple addition statement aloud (e.g., 1 + 1 = 2), it’s much harder to read other equations aloud (e.g., Maxwell’s equations). Also, the spoken statements would be rendered in the speaker’s native language, not a universal tongue.

However, sign language would also be disqualified based on this criterion. Most linguists accept sign language as a true language.

Key Points

  • In order to be considered a language, a system of communication must have vocabulary, grammar, syntax, and people who use and understand it.
  • Mathematics meets this definition of a language. Linguists who don’t consider math a language cite its use as a written rather than spoken form of communication.
  • Math is a universal language. The symbols and organization to form equations are the same in every country of the world.
Religion vs Science

Religion vs Science

As the West becomes more and more secular, and the discoveries of evolutionary biology and cosmology shrink the boundaries of faith, the claims that science and religion are compatible grow louder. If you’re a believer who doesn’t want to seem anti-science, what can you do? You must argue that your faith – or any faith – is perfectly compatible with science.

And so one sees claim after claim from believers, religious scientists, prestigious science organisations and even atheists asserting not only that science and religion are compatible, but also that they can actually help each other. This claim is called “accommodationism.”

But I argue that this is misguided: that science and religion are not only in conflict – even at “war” – but also represent incompatible ways of viewing the world.

My argument runs like this. I’ll construe “science” as the set of tools we use to find truth about the universe, with the understanding that these truths are provisional rather than absolute. These tools include observing nature, framing and testing hypotheses, trying your hardest to prove that your hypothesis is wrong to test your confidence that it’s right, doing experiments and above all replicating your and others’ results to increase confidence in your inference.

And I’ll define religion as does philosopher Daniel Dennett: “Social systems whose participants avow belief in a supernatural agent or agents whose approval is to be sought.” Of course many religions don’t fit that definition, but the ones whose compatibility with science is touted most often – the Abrahamic faiths of Judaism, Christianity and Islam – fill the bill.

Next, realize that both religion and science rest on “truth statements” about the universe – claims about reality. The edifice of religion differs from science by additionally dealing with morality, purpose and meaning, but even those areas rest on a foundation of empirical claims. You can hardly call yourself a Christian if you don’t believe in the Resurrection of Christ, a Muslim if you don’t believe the angel Gabriel dictated the Qur’an to Muhammad, or a Mormon if you don’t believe that the angel Moroni showed Joseph Smith the golden plates that became the Book of Mormon. After all, why accept a faith’s authoritative teachings if you reject its truth claims?

Indeed, even the Bible notes this: “But if there be no resurrection of the dead, then is Christ not risen: And if Christ be not risen, then is our preaching vain, and your faith is also vain.”

Many theologians emphasize religion’s empirical foundations, agreeing with the physicist and Anglican priest John Polkinghorne:

“The question of truth is as central to [religion’s] concern as it is in science. Religious belief can guide one in life or strengthen one at the approach of death, but unless it is actually true it can do neither of these things and so would amount to no more than an illusory exercise in comforting fantasy.”

The conflict between science and faith, then, rests on the methods they use to decide what is true, and what truths result: These are conflicts of both methodology and outcome.

In contrast to the methods of science, religion adjudicates truth not empirically, but via dogma, scripture and authority – in other words, through faith, defined in Hebrews 11 as “the substance of things hoped for, the evidence of things not seen.” In science, faith without evidence is a vice, while in religion it’s a virtue. Recall what Jesus said to “doubting Thomas,” who insisted in poking his fingers into the resurrected Savior’s wounds: “Thomas, because thou hast seen me, thou hast believed: blessed are they that have not seen, and yet have believed.”

And yet, without supporting evidence, Americans believe a number of religious claims: 74 percent of us believe in God, 68 percent in the divinity of Jesus, 68 percent in Heaven, 57 percent in the virgin birth, and 58 percent in the Devil and Hell. Why do they think these are true? Faith.

But different religions make different – and often conflicting – claims, and there’s no way to judge which claims are right. There are over 4,000 religions on this planet, and their “truths” are quite different. (Muslims and Jews, for instance, absolutely reject the Christian belief that Jesus was the son of God.) Indeed, new sects often arise when some believers reject what others see as true. Lutherans split over the truth of evolution, while Unitarians rejected other Protestants’ belief that Jesus was part of God.

And while science has had success after success in understanding the universe, the “method” of using faith has led to no proof of the divine. How many gods are there? What are their natures and moral creeds? Is there an afterlife? Why is there moral and physical evil? There is no one answer to any of these questions. All is mystery, for all rests on faith.

The “war” between science and religion, then, is a conflict about whether you have good reasons for believing what you do: whether you see faith as a vice or a virtue.

Compartmentalizing realms is irrational

So how do the faithful reconcile science and religion? Often they point to the existence of religious scientists, like NIH Director Francis Collins, or to the many religious people who accept science. But I’d argue that this is compartmentalization, not compatibility, for how can you reject the divine in your laboratory but accept that the wine you sip on Sunday is the blood of Jesus?

Others argue that in the past religion promoted science and inspired questions about the universe. But in the past every Westerner was religious, and it’s debatable whether, in the long run, the progress of science has been promoted by religion. Certainly evolutionary biology, my own field, has been held back strongly by creationism, which arises solely from religion.

What is not disputable is that today science is practiced as an atheistic discipline – and largely by atheists. There’s a huge disparity in religiosity between American scientists and Americans as a whole: 64 percent of our elite scientists are atheists or agnostics, compared to only 6 percent of the general population – more than a tenfold difference. Whether this reflects differential attraction of nonbelievers to science or science eroding belief – I suspect both factors operate – the figures are prima facie evidence for a science-religion conflict.

The most common accommodationist argument is Stephen Jay Gould’s thesis of “non-overlapping magisteria.” Religion and science, he argued, don’t conflict because: “Science tries to document the factual character of the natural world, and to develop theories that coordinate and explain these facts. Religion, on the other hand, operates in the equally important, but utterly different, realm of human purposes, meanings and values – subjects that the factual domain of science might illuminate, but can never resolve.”

This fails on both ends. First, religion certainly makes claims about “the factual character of the universe.” In fact, the biggest opponents of non-overlapping magisteria are believers and theologians, many of whom reject the idea that Abrahamic religions are “empty of any claims to historical or scientific facts.”

Nor is religion the sole bailiwick of “purposes, meanings and values,” which of course differ among faiths. There’s a long and distinguished history of philosophy and ethics – extending from Plato, Hume and Kant up to Peter Singer, Derek Parfit and John Rawls in our day – that relies on reason rather than faith as a fount of morality. All serious ethical philosophy is secular ethical philosophy.

In the end, it’s irrational to decide what’s true in your daily life using empirical evidence, but then rely on wishful-thinking and ancient superstitions to judge the “truths” undergirding your faith. This leads to a mind (no matter how scientifically renowned) at war with itself, producing the cognitive dissonance that prompts accommodationism. If you decide to have good reasons for holding any beliefs, then you must choose between faith and reason. And as facts become increasingly important for the welfare of our species and our planet, people should see faith for what it is: not a virtue but a defect.

Brexit: A Profound level of political stalemate

Brexit: A Profound level of political stalemate

After the collapse of Theresa May’s efforts to push her withdrawal agreement through parliament, and the failed subsequent Conservative Party motion of no-confidence against her, two factors have been fairly well established. There appears to be no withdrawal agreement available that commands a majority in the House of Commons, least of all the current one. And both parliament and the Conservative Party lack the will to oust the prime minister. Not only is opposition leader Jeremy Corbyn reluctant to submit the motion required for parliament to hold a vote of no confidence because he does not think it would pass, but in addition, May is safe from another confidence vote from her party for a full year.

In short, Britain is now in a situation where the prime minister is relatively safe in her position but unable to move forward on the most important subject of the day. This is a profound level of political stalemate at a time when some form of forward motion is urgently required.

There are no immediate solutions available. The problems that led to this impasse run deep. However, two medium to long-term measures would at least help alleviate the risk of future stalemate and crisis.

One is reforming the way the UK elects its members of parliament. First past the post (FPTP, or “single member plurality”, as is the technical name) has a number of problems. Perhaps most damagingly, the distribution of votes is very poorly reflected in the distribution of seats. In 2015 UKIP scored an all-time high result of 3,881,099 votes, or 12.6%. Its parliamentary reward was one seat. In comparison, the SNP received 1,454,436 votes, or 4.7%. Its reward was 56 seats.

One SNP seat cost on average 25,972 votes, compared to, by definition, 3,881,099 votes for UKIP, or 149 times more than SNP seats. If those who have concerns about the UK’s international partnerships see their voices so blatantly ignored in general elections, it is perhaps not surprising that they welcomed the opportunity to vote Leave when the chance appeared. Although the UK did hold a referendum on electoral reform in 2011, and rejected it, the proposed Alternative Vote system would not have dealt with the problem of some parties gaining vastly more for their votes than others.

The current system is also forcing the two main parties to maintain what are blatantly very unhappy and dysfunctional political family units. Labour and the Conservatives have to contain within them people whose views diverge radically from each other. The sheer depth of the problem was painfully illustrated on the day of the confidence vote when two Conservative MPs apparently refused to feature in a TV interview together, live on air.

One Conservative MP described the situation in his party as something akin to Mad Max or Lord of the Flies. This situation is leading to damaging struggles over control of the leadership and wider organisation as different fractions vie for control. Labour’s internal battles over Corbyn as leader and the role of the grassroots Momentum group are further illustrations of this problem.

It seems clear that what is needed is a system of electing MPs that better reflects how people actually vote. This would help to channel concerns and objections about the political direction of the country in more meaningful and constructive ways to avoid festering dissent building up. It would also help split up the internally squabbling and dysfunctional parties, creating organisations more at peace with themselves ideologically.

A recent Ashcroft poll showed that when respondents were asked if they thought May’s Brexit deal honoured the result of the referendum and whether it was better than leaving without a deal or remaining, the answer “don’t know” was often the either first or second most popular choice.

If we are going to use referendums as a means to settle big political questions (and ideally we should not) then clearly the population needs to be sufficiently knowledgeable to make informed decisions. If citizens are to have healthy debates on future big decisions and electoral choices, political literacy needs to be given a much higher priority, both in formal education and beyond.

A recent House of Lord’s report stated that “citizenship education has a crucial role to play in helping to build active citizens”. It seems profoundly unwise to use referendums to answer big political questions if voters are not equipped to take such active role in decision making as a referendum allows. In addition, the same report noted that citizenship education, and arguably political literacy more generally, is central to encouraging “a thoughtful national narrative about Britishness”.

If the prime minister wants to succeed in bringing the country together post Brexit, which was a central theme in her speech to the Conservative Party conference, a “thoughtful narrative” about the political system that exacerbated this Brexit debacle seems a good place to start.

Ebola outbreak in the Democratic Republic of the Congo

Ebola outbreak in the Democratic Republic of the Congo

The Ebola outbreak in the Democratic Republic of the Congo (DRC) is quickly becoming an international concern. With 489 people diagnosed with the disease and 280 deaths, it is already the second largest Ebola outbreak in history (although still dwarfed by the 2014 West Africa outbreak). While the DRC has a good track record of responding to these outbreaks, the ongoing military conflict in the country is making the response much more difficult.

Healthcare workers have been attacked, vaccination campaigns halted and, most recently, experts from the US Centers for Disease Control and Prevention were recalled from an outbreak zone due to security fears. This is all despite the DRC hosting the UN’s largest peacekeeping force, MONUSCO (Mission de l’Organisation des Nations unies pour la stabilisation en République démocratique du Congo).

The United Nations Security Council (UNSC), the body responsible for maintaining international peace and security, acknowledged the situation in the DRC by issuing a resolution (resolution 2439) that calls for an end to the violence so that humanitarian groups can have better access to the region.

This is not the first time the UNSC has been involved in an Ebola outbreak. In 2014 it passed a resolution on the West African Ebola outbreak. This was the first and only UNSC resolution to acknowledge a health issue as a threat to peace and security. The resolution didn’t lead to military intervention, but it greatly improved the response in West Africa. The UNSC asked for resources and action, and the international community respondedpromptly.

This time, though, the UNSC response is much weaker. It condemns the killing of healthcare workers, calls for peace on both sides (which has been ignored) and calls on the World Health Organisation (WHO) to continue its leadership on Ebola, but does not say much else. With the violence continuing, the number of cases and deaths from Ebola rising, and the withdrawal of international health experts, it is likely that the UNSC will need to return to this issue – this time with a more robust response. But what form might this response take?

No easy options

One option is for the UNSC to do nothing. It could view the conflict as the responsibility of the DRC government, Ebola as the responsibility of the WHO and partner health agencies, and the mandate of MONUSCO being limited to traditional peacekeeping operations. Given the escalating situation, though, this option seems unlikely.

Another option is for the UNSC to draft another resolution, this time clearly linking the conflict in the DRC and Ebola. It would use stronger language than the last resolution to encourage an end to fighting and inspire the international response necessary to end the Ebola outbreak. A stronger resolution would reiterate the importance of the issue and draw international attention and resources to Ebola. However, without peace and security being assured, the international community is unlikely to send resources and people to the affected areas.

A more likely option appears to be expanding the role of the MONUSCO peacekeepers to take a more active role in protecting healthcare workers in the DRC so that healthcare workers can safely carry out their work. While the UNSC has moved away from authorising military-led peacekeeping missions, the fact that there is a peacekeeping force already in the DRC means an expansion of their mandate is more likely.

In particular, MONUSCO could create and maintain a corridor of safety when combatants are only engaged by peacekeepers if they enter this area. The UNSC created such a zone in the Iraq-Kuwait conflict as a form of humanitarian intervention so that water and aid could be delivered to civilians within the corridor.

Creating a safety zone around the Ebola outbreaks could allow healthcare professionals to carry out their work in safety and may inspire international assistance to flow into the area where it is so badly needed. While mission creep within peacekeeping operations is rightly subject to considerable criticism, if the situation in the DRC worsens, it may be the only option.

I Am Cait, reigniting the internet debate

I Am Cait, reigniting the internet debate

On July 26, Caitlyn Jenner premiered her new reality show I Am Cait, reigniting the internet debate on the validity of the transgender identity. Which side of the debate one falls on correlates highly with one’s political position.

Liberal politicians like President Barack Obama tend to support Jenner and commend her on her courage.

On the other hand, conservative politicians like Mike Huckabee have mocked Jenner’s transition, some of them referring to her as mentally ill.

Of course, there are exceptions. Some conservatives support transgender individuals – indeed, Jenner herself identifies as a Republican – and some liberals deny that Jenner is a woman. But we think the conservative–liberal divide is prevalent enough to be worthy of attention.

Getting the facts straight

Those on the left and right seem to believe that they are motivated by a desire to get to the fact of the matter about what constitutes being a man or a woman. That is, they think that they are arguing for an unbiased account about what gender is.

One way they do this is by referring to anatomy. Many on the left argue that gender is “deeply rooted in one’s mind.” They cite psychologists like Columbia’s Derald Wing Sue, who argues that “Caitlyn Jenner is not identifying with being a woman because of (her) upbringing and cultural conditioning”; rather, her gender is biologically programmed into her.

Those on the right often argue that being a man or woman is simply being born with male or female sex organs or that people who are transgender are mentally ill. They cite studies showing that individuals who have undergone reassignment surgery are more likely to commit suicide than those who have not. The increased risk of suicide is thought to show that identifying as transgender is a consequence of depression and, therefore, not a genuine identity.

An intense debate

The debate around these issues is intense. Controversial writer (and cofounder of Vice) Gavin McInnes’ article “Transphobia Is Perfectly Natural”, for example, has elicited over 5,000 comments.

As empirically oriented philosophers with research interests in what motivates individuals’ reasoning, we suspect that the debate is not motivated by a desire to get to the fact of the matter about gender. After all, when individuals disagree on other facts, such as (say) whether Napoleon won at Waterloo, we cannot predict people’s beliefs based on their political affiliation. Our view is that the intensity of this discussion is best explained by what psychologists call identity protective cognition.

Identity protective cognition is the tendency to selectively accept and dismiss information to support one’s identity.

This theory, developed by Dan M Kahan of Yale Law School, argues that beliefs about how society “ought to be” are central to one’s group identity. People discount information if it suggests their group’s picture of the ideal society is lacking.

For conservatives, the ideal society tends to be hierarchical. They want – perhaps subconsciously – resource distribution to depend on factors such as social class, race and sex. Status flows to men who work at well-paying jobs and women who stay at home and tend to the family.

Transgender individuals are dangerous to the hierarchy conservatives desire. They complicate the binary picture of division of labor within the traditional family. The existence of transgender individuals suggests that the assumption that underpins the legitimacy of the hierarchy – that people designated men and women at birth are naturally suited to particular gender roles – is false. So conservatives are motivated to deny the reality of transgenderism.

On the opposite side of things, research has shown that liberals tend to favor an egalitariansociety. They want resources to be divided more equally, and they do not want the division to depend on gender. Social status should not be determined by conformity to gender stereotypes.

In such a society, transgender identities must be legitimate. Otherwise, there is a risk of propagating the view that one has to fall neatly into one of two genders, a view that forces men and women into unequal social roles.

We believe that if the debate at hand is to make real progress, we need to recognize that it is not merely about whether Jenner is female. It is implicitly a debate about how we ought to structure society and people’s roles in it. It is a debate not just about what certain words mean; it is about what they ought to mean.

The holiness of God

The holiness of God is one of his attributes that carries monumental consequences for every person on earth.

In ancient Hebrew, the word translated as “holy” (qodeish) meant “set apart” or “separate from.” God’s absolute moral and ethical purity set him apart from every other being in the universe.

The Bible says, “There is no one holy like the Lord.” (1 Samuel 2:2, NIV)

The prophet Isaiah saw a vision of God in which seraphim, winged heavenly beings, called to each other, “Holy, holy, holy is the Lord Almighty.” (Isaiah 6:3, NIV) The use of “holy” three times stresses God’s unique holiness, but some Bible scholars also believe there is one “holy” for each member of the Trinity: God the Father, Son, and Holy Spirit. Each Person of the Godhead is equal in holiness to the others.

For human beings, holiness generally means obeying God’s law, but for God, the law is not external—it is part of his essence. God is the law. He is incapable of contradicting himself because moral goodness is his very nature.

God’s Holiness Is a Recurring Theme in the Bible

Throughout Scripture, the holiness of God is a recurring theme. The Bible writers draw a sharp contrast between the Lord’s character and that of humankind. God’s sacredness was so high that writers of the Old Testament even avoided using the personal name of God, which God revealed to Moses from the burning bush on Mount Sinai.

The earliest patriarchs, Abraham, Isaac, and Jacob, had referred to God as “El Shaddai,” meaning The Almighty. When God told Moses his name is “I AM WHO I AM,” translated as YAHWEH in Hebrew, it revealed him as the Uncreated Being, the Self-Existing One. Ancient Jews considered that name so holy they would not pronounce it aloud, substituting “Lord” instead.

When God gave Moses the Ten Commandments, he expressly forbid using the name of God disrespectfully. An attack on God’s name was an attack on God’s holiness, a matter of grave contempt.

Ignoring God’s holiness brought deadly consequences. Aaron’s sons Nadab and Abihu, acted contrary to God’s commands in their priestly duties and he killed them with fire. Many years later, when King David was having the ark of the covenant moved on a cart—in violation of God’s commands—it tipped when the oxen stumbled, and a man named Uzzah touched it to steady it. God immediately struck Uzzah dead.

The Holiness of God Is the Basis for Salvation

Ironically, the plan of salvation was based on the very thing that separated the Lord from mankind: the holiness of God. For hundreds of years, the Old Testament people of Israel were bound to a system of animal sacrifices to atone for their sins. However, that solution was only temporary. As far back as Adam, God had promised the people a Messiah.

A Savior was necessary for three reasons. First, God knew human beings could never meet his standards of perfect holiness by their own behavior or good works. Second, he required a spotless sacrifice to pay the debt for humanity’s sins. And third, God would use Messiah to transfer holiness to sinful men and women.

To satisfy his need for a faultless sacrifice, God himself had to become that Savior. Jesus, the Son of God, was incarnated as a human being, born of a woman but retaining his holiness because he was conceived by the power of the Holy Spirit. That virgin birth prevented the passing of Adam’s sin on to the Christ child. When Jesus died on the cross, he became the fitting sacrifice, punished for all the sins of the human race, past, present, and future.

God the Father raised Jesus from the dead to show that he accepted Christ’s perfect offering. Then to guarantee humans meet his standards, God imputes, or credits Christ’s holiness to every person who receives Jesus as Savior. This free gift, called grace, justifies or makes holy every Christ follower. Bearing Jesus’ righteousness, they are then qualified to enter heaven.

But none of this would have been possible without God’s tremendous love, another of his perfect attributes. Through love God believed the world was worth saving. That same love led him to sacrifice his beloved Son, then apply Christ’s righteousness to redeemed human beings. Because of love, the very holiness that seemed to be an insurmountable obstacle became God’s way to grant eternal life to everyone who seeks him.

How mathematicians still grapple with the issues Einstein confronted

Albert Einstein released his general theory of relativity at the end of 1915. He should have finished it two years earlier. When scholars look at his notebooks from the period, they see the completed equations, minus just a detail or two. “That really should have been the final theory,” said John Norton, an Einstein expert and a historian of science at the University of Pittsburgh.

But Einstein made a critical last-second error that set him on an odyssey of doubt and discovery — one that nearly cost him his greatest scientific achievement. The consequences of his decision continue to reverberate in math and physics today.

Here’s the error. General relativity was meant to supplant Newtonian gravity. This meant it had to explain all the same physical phenomena Newton’s equations could, plus other phenomena that Newton’s equations couldn’t. Yet in mid-1913, Einstein convinced himself, incorrectly, that his new theory couldn’t account for scenarios where the force of gravity was weak — scenarios that Newtonian gravity handled well. “In retrospect, this is just a bizarre mistake,” said Norton.

To correct this perceived flaw, Einstein thought he had to abandon what had been one of the central features of his emerging theory.

Einstein’s field equations — the equations of general relativity — describe how the shape of space-time evolves in response to the presence of matter and energy. To describe that evolution, you need to impose on space-time a coordinate system — like lines of latitude and longitude — that tells you which points are where.

The most important thing to recognize about coordinate systems is that they’re human contrivances. Maybe in one coordinate system we label a point (0, 0, 0), and in another we label that same point (1, 1, 1). The physical properties haven’t changed — we’ve just tagged the point differently. “Those labels are something about us, not something about the world,” said James Weatherall, a philosopher of science at the University of California, Irvine.

Einstein initially wanted his equations to be coordinate-independent (a property he called “general covariance”), meaning they’d produce correct, consistent descriptions of the universe regardless of which coordinate system you happened to be using. But Einstein convinced himself that in order to fix the error he thought he’d made, he had to abandon general covariance.

Not only did he fail at this, he doubled down on his error: He tried to show that coordinate independence was not a property that his theory could have, even in principle, because it would violate the laws of cause and effect. As one study of Einstein put it, “Nothing is easier for a first-rate mind than to form plausible arguments that what it cannot do cannot be done.”

Einstein pulled out of this dive just in time. By late 1915 he knew that the influential German mathematician David Hilbert was close to finalizing a theory of general relativity himself. In a few feverish weeks in November 1915, Einstein reverted to the equations of general relativity he’d had in hand for more than two years and applied the finishing touches. In November 1915, in the first of four lectures before the Prussian Academy of Sciences, he announced his achievement. Our view of the physical world has not been the same since.

The Einstein field equations we have today are generally covariant. They express the same physical truths about the universe — how space-time curves in the presence of energy and matter — regardless of what coordinates you use to label things.

Yet mathematicians and physicists still grapple with the issues around coordinate systems that slowed Einstein a century ago. For example, the monumental effort to reconcile general relativity with quantum theory flounders in part because of the difficulty of developing a theory of quantum gravity that has the same general covariance Einstein achieved with his field equations. “In some sense you could argue the reason we don’t have an adequate quantum theory of gravity is we don’t know how to express the solutions to Einstein’s equations in a way that completely removes any kind of coordinate dependence,” said Weatherall.

In practice, the challenge is often figuring out how to break the general covariance of Einstein’s equations — that is, how to choose a specific coordinate system that is well suited to solving a specific problem. The issue has proven especially acute for mathematicians who study the so-called black hole stability conjecture, which I wrote about in my recent article “To Test Einstein’s Equations, Poke a Black Hole.” Depending on the particular problem you’re interested in, some coordinate systems work better than others — and figuring out which coordinate system to choose, and how to adjust it as the solution changes, is a high mathematical art.

New proofs would come much easier if there were a single universal coordinate system that worked for every problem and every configuration of space-time. But as Einstein discovered during those fraught, wandering years, the universe doesn’t admit any one privileged choice of coordinates.

“It’s not just that we don’t have such a choice,” said Weatherall. “It’s that one of the things we take Einstein to have taught us is that it would be a mistake to expect there to be such a choice.”

The Third Revolution

“One of the great paradoxes of China today,” writes eminent China scholar Elizabeth C. Economy, “is Xi Jinping’s effort to position himself as a champion of globalization, while at the same time restricting the free flow of capital, information, and goods between China and the rest of the world.” In her new book, The Third Revolution: Xi Jinping and the New Chinese State, Economy explains that “the ultimate objective of Xi’s revolution is his Chinese Dream—the rejuvenation of the great Chinese nation.”

Characterized by “a reassertion of the state in Chinese political and economic life at home, and a more ambitious and expansive role for China abroad,” Xi’s China is exercising “new levers of influence and power that others will have to learn to exploit and counter to protect and advance their own interests,” warns Economy, C. V. Starr senior fellow and director of Asia Studies at CFR.

Xi has reversed the thirty years of reform and opening initiated by former Chinese leader Deng Xiaoping’s Second Revolution and replaced it with his own Third Revolution, she writes. “What makes Xi’s revolution distinctive is the strategy he has pursued: the dramatic centralization of authority under his personal leadership; the intensified penetration of society by the state; the creation of a virtual wall of regulations and restrictions that more tightly controls the flow of ideas, culture, and capital into and out of the country; and the significant projection of Chinese power.”

“An illiberal state seeking leadership in a liberal world order,” China poses a set of distinct new challenges for the United States. Xi “seeks to project the current Chinese political and economic development model globally,” and to “become a standard bearer for other countries disenchanted with the American and European models of liberal democracy.” While China “takes advantage of the openness of the United States and other market-based liberal democracies to further its economic interests and advance its political and cultural influence,” it “increasingly constrains opportunities” for other countries to do the same.

“The United States . . . must continue to seek opportunities for cooperation but at the same time be prepared to counter and confront China when Xi’s Third Revolution spills over into the rest of the world, undermining the principles underpinning global security and prosperity it purports to uphold,” she writes.

Economy urges the United States to adopt a strategic framework for its relationship with China that establishes U.S. priorities and the diplomatic, economic, and military approaches necessary to realize them, “[retaining] what has worked well for its policy toward China while adapting to a new political reality.” She offers several recommendations to U.S. policymakers:

  • Leverage Xi’s ambition for leadership and “[encourage] China to do more on the global stage,” such as addressing the global refugee crisis and ensuring that its global development strategy, the Belt and Road Initiative, adopts better governance practices.
  • Advance technical cooperation between China and the United States around the big issues of global governance “to build an institutional infrastructure for cooperation.”
  • Work with U.S. allies and partners in Asia and Europe to support the underlying principles of a “free and open Indo-Pacific, rooted in a rules-based order” by revisiting “U.S. participation in the [Trans-Pacific Partnership],” deterring “further efforts by China to realize its sovereignty claims through unilateral actions” in the South China Sea and Taiwan, and developing “programs that build good governance capacity” such as the rule of law, in countries such as Ethiopia and Pakistan, where Chinese political influence is expanding.
  • When U.S. political, economic, or security interests are “directly and meaningfully undermined,” adopt firmer policies such as economic retaliation and reciprocal action, “making clear to China the costs of noncompliance with agreements or established norms.”
  • Prioritize U.S. government support for the “development and adoption of new technologies” to compete effectively with Made in China 2025 and other innovation-driven industrial policies.
  • Support through “both word and deed,” fundamental values including “democracy and respect for human rights, a market economy, and free trade.”

“China cannot be a leader in a globalized world while at the same time closing its borders to ideas, capital, and influences from the outside world,” Economy concludes.

Questions about exploring outer space

Questions about exploring outer space

Metallic shrapnel flying faster than bullets; the Space Shuttle smashed to pieces; astronauts killed or ejected into space. The culprit? Space debris – remnants of a Russian satellite blown up by a Russian missile. The one survivor, Ryan Stone, has to find her way back to Earth with oxygen supplies failing and the nearest viable spacecraft hundreds of miles away.

Over on Mars, 20 years in the future, an exploration mission from Earth is going wrong. An epic dust storm forces the crew to abandon the planet, leaving behind an astronaut, Mark Watney, who is presumed dead. He has to figure out how to grow food while awaiting rescue.

Hollywood knows how to terrify and inspire us about outer space. Movies like Gravity (2013) and The Martian (2015), present space as hostile and unpredictable – spelling danger for any intrepid human who dares to venture outside Earth’s hospitable confines.

This is only part of the story, however – the bit with people centre stage. Sure, no one wants to see astronauts killed or stranded in space. And we all want to enjoy the fruits of successful planetary science, like determining which planets could host human life or simply whether we’re alone in the universe.

Valuing space

But should we care about the universe beyond how it affects us as humans? That is the big question – call it question #1 of extraterrestrial environmental ethics, a field too many people have ignored for too long. I’m one of a group of researchers at the University of St Andrews trying to change that. How we ought to value the universe depends on two other intriguing philosophical questions:

Question #2: the kind of life we are most likely to discover elsewhere is microbial – so how should we view this lifeform? Most people would accept that all humans have intrinsic value, and matter not only in relation to their usefulness to someone else. Accept this and it follows that ethics places limits on how we may treat them and their living spaces.

People are starting to accept that the same is true of mammals, birds and other animals. So what about microbial beings? Some philosophers like Albert Schweitzer and Paul Taylor have previously argued that all living things have a value in themselves, which would obviously include microbes. Philosophy as a whole has not reached a consensus, however, on whether it agrees with this so-called biocentrism.

Question #3: for planets and other places not hospitable to life, what value should we place on their environment? Arguably we care about our environment on Earth primarily because it supports the species that live here. If so, we might extend the same thinking to other planets and moons that can support life.

But this doesn’t work for “dead” planets. Some have proposed an idea called aesthetic value, that certain things should be treasured not because they are useful but because they are aesthetically wonderful. They have applied this not only to great artistic works like Leonardo da Vinci’s Mona Lisa and Beethoven’s Fifth, but also to parts of the Earth’s environment, such as the Grand Canyon. Could that apply to other planets?

Alien environments

Supposing we could answer these theoretical questions, we could proceed to four important practical questions about space exploration:

Question #4: is there a duty to protect the environment on other planets? When it comes to sending astronauts, instruments or robots to other worlds, there are clearly important scientific reasons for making sure they don’t take terrestrial organisms with them and wind up depositing them there.

Otherwise, if we discovered life, we wouldn’t know whether it was indigenous – not to mention the risk of wiping it out entirely. But is scientific clarity all that matters, or do we need to start thinking about galactic environmental protection?

Question #5: what, besides biological contamination, would count as violating such an obligation to treat that planet’s environment with respect? Drilling for core samples, perhaps, or leaving instruments behind, or putting tyre tracks in the dirt?

Question #6: what about asteroids? The race is well underway to develop technology to harvest the untold trillions of pounds of mineral wealth presumed to exist on asteroids, as already reported in The Conversation. It helps that no one seems to think of asteroids as environments we need to protect.

The same goes for empty space. The movie Gravity gave us some human-centred reasons to be worried about the buildup of debris in space, but might there be other reasons to object? If so, would our obligation be to merely create less debris, or something stronger – like not producing any new debris or even cleaning up what we’ve left already?

Question #7: what considerations might offset arguments in favour of behaving ethically in space? Of the various reasons for going there – intellectual/scientific, utilitarian, profit-driven – are any strong enough to override our obligations?

We also need to factor in the inevitable risks and uncertainties here. We can’t know what benefits space missions will have. We can’t be certain of not biologically contaminating the planets we visit. What risk/reward trade-offs should we be willing to undertake?


Discussions about outer space have the advantage that we have very little attachment to anything out there. These ethical questions might therefore be some of the only ones humans can address with a large measure of emotional distance. For this reason, answering them might help us to make progress with Earth-bound issues like global warming, mass extinction and nuclear waste disposal.

Space exploration also directly raises questions about our relationship to Earth – once we overcome the technological puzzles preventing the terraforming of a planet like Mars, or find ways of reaching habitable exoplanets. I’ll leave you with one extremely important one for the future:

Question #8: given that the Earth is not the only potential home for human beings, what reasons for protecting its environment would remain once we can realistically go somewhere else?

Some Eastern European youngsters living in the UK say they feel less welcome since Brexit​

It is hard for me to identify as both British and Romanian because people make me feel as if you can’t be both – as if being foreign is a permanent thing and can’t be changed no matter how long you’ve spent in a country or what you consider to be “home”. I consider the UK to be my home.

Like Romanian teenager Ioana, whose words above articulate her painful reality, 15-year-old Alicja moved to Britain from Bulgaria when she was a young child. On the night of the EU referendum, her family gathered around the TV to watch the result. Her mother said she saw it coming; having listened to comments at work about Eastern Europeans taking local jobs in their small fishing town, she realised that anti-immigration feelings were running deep. For Alicja, who had grown up in Scotland, “Brexit had me in tears – it has changed everything”.

In the months since, her family’s economic security and plans to stay in the UK are up in the air. They don’t have the money to apply for citizenship (currently £1,330 per person) and they don’t even know if they would qualify, with Alicja’s mother in part-time work. But what is clear is the impact that Brexit has had on young people’s sense of belonging in Britain.

No future?

Young Europeans living in the UK have been considerably affected by the decision to leave the European Union, underscored by the rise in applications for British citizenship from EU citizens and the recorded increase in migration of EU citizens from the UK since the referendum.

Our research project is the largest study of Eastern European young people aged 12 to 18 living in the UK, since the EU referendum. Like Alicja, the majority of the survey participants said they felt “uncertain” (56%), and “worried” (54%), while just over a quarter (27%) said they were “scared” about their future.

Although most had lived in the UK for more than five years, only 8% had British or dual nationality. While the UK government has promised to make applications for settled statusstraightforward for EU nationals, there is evidence that many groups – including children in vulnerable families, children in care or those with parents in insecure work, are at risk of becoming undocumented. Young people in our study had an acute sense of insecurity about their future. As Polish-born Renata, 15, said:

I will need to get citizenship in the UK to stay. I’m still not sure what my parents will do, we definitely can’t afford for all of us to get citizenship, so they might have to move back, while I’ll need to live by myself here.

Like the children of the Windrush generation, young Eastern Europeans arrived in the UK mainly because of their parents’ desire for a more secure future. But for many, growing up in “austerity Britain” has meant an increasing sense of feeling unwelcome, given the growing hostile attitude to immigration.

Prejudice and pride

Such an environment affects the everyday lives of young Eastern Europeans living in the UK and does not help integration. More than three quarters (77%) of our participants said they have experienced racism and xenophobia, and for one in five, these experiences happened “often” in school. A third also thought that their neighbours had some level of prejudice against Eastern Europeans, which made some feel unsafe and worried they might be attacked. Many said they adopted “blending in” tactics, like not speaking their own language in public or putting on a local accent.

So how do you develop a sense of belonging in a place where you generally feel unwanted? And what effect does it have on your sense of identity, especially in these formative years?

As many had strong links to Europe through birthplace and regular visits, it is unsurprising that 92% said they felt European. They had a strong sense of connection and belonging to Europe, with many saying that a European identity would always be part of who they were and how they saw their place in the world.

Anchoring themselves in a European identity seemed to offer security during insecure times in Britain. However, the majority (83%) felt they belonged in the UK, and this feeling became stronger the longer young people had lived here. Less than half (41%) said they felt British. Navigating these identities – in addition to other dimensions such as gender, class and religion – is clearly a complex and emotionally charged process for many. As Polish-born Emilia, 16, put it:

I may live in the UK, but I’ve been brought up in a Polish house. There’s still a part of me that doesn’t feel fully connected. I also feel like a fake Pole – like I’m not really part of that culture either. I’m stuck in the middle, just doing my best to fit in with whoever will let me.

Three quarters said they are likely to stay, many with plans to continue their education, volunteering or work, while others consider leaving and planning a future elsewhere. Many of them are clear there is no “going back” to the country of their birth, but rather envisage their future elsewhere. Latvia-born Michael, 18, said:

I feel very connected to Europe and European culture. There has been some concern whether I want to stay here due to the political changes. I might move to the EU after finishing university, despite the fact that I enjoy living in this country.

Many EU-born young people who arrived in Britain when their parents migrated are clearly emotionally bruised by the UK’s decision to leave the EU, which for many was like “a kick in the teeth”. Educated in Britain, they are now at the point of making decisions about their own futures – but here or elsewhere? For most, Britain is their home, with connections to family, friends, places and memories.

Migrating elsewhere will not be easy and while some will stay, others are increasingly looking beyond Britain for their future. For employers, educators and policy makers, one of the key questions now is what this country needs to do to encourage young Europeans to stay.