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enWho's watching: Can maths exist if you can't see it?
https://plus.maths.org/content/whos-watching-can-maths-exist-if-you-cant-see-it
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/constructive_icon_1.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In our <a href="/content/whos-watching-physics-observers"><em>Who's watching</em> project</a> we have been exploring the physics of observers. But observations also play a role in mathematics. We often get ideas about general results in mathematics by observing specific examples and specific examples also give us a feel for what
a general result really means.</p>
<p>But unlike physicists, mathematicians don't need to "see" something to be certain it exists. Many mathematical proofs show that a mathematical object exists by logical necessity, without actually constructing it. Are these proofs really as valid as constructive proofs? And what happens if you try and avoid them? Find out with this collection of articles.</p>
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<p><a href="/content/something-nothing">Something from nothing?</a> — If you can prove that a statement can't possibly be false, does this mean it's true? This article explores the law of the excluded middle and how it is used in mathematics.</p></div>
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<p><a href="/content/intuitionism">Intuitive mathematics</a> — Can a mathematical object be said to exist if you can't construct it? This article looks at a school of thought called intuitionism, which rejects the law of the excluded middle and non-constructive proofs.</p></div>
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<p><a href="/content/paradise">Paradise lost and rescued</a> — What kind of mathematics can you do if you insist that everything must be constructed? This article, the second part of the one above, explores constructivist mathematics.</p></div> <br clear="all" />
<h3>Background reading and further reading</h3>
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<p><a href="/content/constructive-mathematics">Constructive mathematics</a> — This is one of our older articles. It also looks at constructivism, what inspired it, and where it can lead to.</p></div>
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<p><a href="/content/maths-minute-real-numbers-and-cauchy-sequences">The real numbers and Cauchy sequences</a> — The real numbers, with their infinite and slippery nature, are a particular concern of constructivism. This article looks at how to define them in terms of infinite sequences of rational numbers.</p></div>
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<p><a href="/content/maths-minute-truth-tables">Maths in a minute: Truth tables</a> — A quick look at those indispensable tools of binary logic and how the law of the excluded middle fits into it.</p></div>
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<p><a href="/content/maths-minute-chomp">Maths in a minute: Chomp</a> — A great example of a non-constructive proof involves a game of maths and biscuits.</p></div>
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<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p><em>This package is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-physics-observers">here</a> to see more articles and videos about questions to do with observers in physics. </em></p></div></div></div>Wed, 18 Jul 2018 14:26:20 +0000Marianne7054 at https://plus.maths.org/contenthttps://plus.maths.org/content/whos-watching-can-maths-exist-if-you-cant-see-it#commentsIntuitive mathematics
https://plus.maths.org/content/intuitionism
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/clock_icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>If you find it hard to live by your principles, then consider the plight of the mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Brouwer.html">Luitzen Egbertus Jan Brouwer</a> at the beginning of the twentieth century. Brouwer had discovered he didn't agree with the way mathematics was being done. To make things right, much of mathematics would have to be rewritten from scratch. It would have been easy to keep quiet, but Brouwer rose to the challenge. His effort didn't gain him much popularity at the time and his ideas never did change mainstream maths.
But they seeded an interesting movement which is still alive today: <em>constructivism</em>.</p>
<h3>What is maths?</h3>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/constructivism/bruwer_icm.png" alt="LEJ Brouwer" width="350" height="261" />
<p>LEJ Brouwer, third from the left, at the <a href="https://en.wikipedia.org/wiki/International_Congress_of_Mathematicians">International Congress of Mathematicians</a> in Zürich in 1932.</p>
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<p>Brouwer's misgivings rested on his view on where mathematics comes from. Many mathematicians of the time (and of today) thought that mathematics exists independently of humans in some kind of <em><a href="https://en.wikipedia.org/wiki/Platonism">Platonic</a></em> realm of eternal truth, which we venture out to explore with our minds. Another prominent school of the time, the <em><a href="https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics)">formalists</a></em>, had stripped
mathematics of all intuition and turned it into a game of pure logic, which, so Brouwer thought, was devoid of meaning. </p>
<p>In Brouwer's view maths was neither independent of us, nor an empty game whose rules we can change at will. To him maths was a human creation rooted in our intuition. Our perception of the passage of time, he thought, with one moment following on from the one before, shapes our intuition of the natural numbers 1, 2, 3, etc, and of the infinity they move towards. Along with other notable mathematicians, Brouwer believed that the natural numbers formed the basis of all the mathematics we have created. Therefore, all mathematical objects and arguments should be regarded as mental constructions we build in the privacy of our own minds, based on our intuition. Logic, according to Brouwer, wasn't the essence of these private ideas, but it's what we use, together with language, to communicate about them. </p>
<h3>Non-constructive proofs</h3>
<p>Esoteric as it may seem, Brouwer's <em>intuitionist</em> stance has stark consequences for everyday mathematics. If mathematical objects are mental constructions, then the only way of proving that a particular mathematical object exists is to give a recipe for mentally constructing it — since, according to Brouwer, "outside human thought there are no mathematical truths", if you can't construct an object in your mind, then it lacks reality. Many mathematical proofs, however, do no such thing: they show that something exists by logical necessity without telling you how to find it.</p>
<p>Such <em>non-constructive</em> proofs have a touch of magic about them,
but can also leave you feeling cheated. A particularly slick example is a proof which shows that there are two irrational numbers <img src="/MI/d98b686119645d3e73909534286c40f9/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/d98b686119645d3e73909534286c40f9/images/img-0002.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" />, so that</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0001.png" alt="\[ a^ b \]" style="width:15px;
height:15px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> is a rational number. You start by considering the number </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0002.png" alt="\[ \sqrt {2}^\sqrt {2} \]" style="width:39px;
height:24px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>If this is a rational number, then set <img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0003.png" alt="$a=\sqrt {2}$" style="vertical-align:-2px;
width:53px;
height:17px" class="math gen" /> and <img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0004.png" alt="$b=\sqrt {2}$" style="vertical-align:-2px;
width:52px;
height:17px" class="math gen" /> and (because you know that <img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0005.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> is irrational) you’re done. If, however, <img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0006.png" alt="$\sqrt {2}^\sqrt {2}$" style="vertical-align:-2px;
width:39px;
height:24px" class="math gen" /> is an irrational number, set <img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0007.png" alt="$a=\sqrt {2}^\sqrt {2}$" style="vertical-align:-2px;
width:71px;
height:24px" class="math gen" /> and <img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0004.png" alt="$b=\sqrt {2}$" style="vertical-align:-2px;
width:52px;
height:17px" class="math gen" />. Then </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/1f08a65c4a8ea73f4c10552886d8f42d/images/img-0008.png" alt="\[ a^ b = \left(\sqrt {2}^{\sqrt {2}}\right)^{\sqrt {2}} =\sqrt {2}^{\sqrt {2} \times \sqrt {2}}= \sqrt {2}^2 = 2, \]" style="width:297px;
height:47px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> which is rational, as required. </p>
For all its elegance, this proof doesn't give you a way of telling which pair of numbers — <img src="/MI/a85ec1d70c68f3f21ce6a17e86cb409a/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> and <img src="/MI/a85ec1d70c68f3f21ce6a17e86cb409a/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" />, or <img src="/MI/a85ec1d70c68f3f21ce6a17e86cb409a/images/img-0002.png" alt="$\sqrt {2}^\sqrt {2}$" style="vertical-align:-2px;
width:39px;
height:24px" class="math gen" /> and <img src="/MI/a85ec1d70c68f3f21ce6a17e86cb409a/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> — actually have the required property. If your life depended on knowing the values of <img src="/MI/a85ec1d70c68f3f21ce6a17e86cb409a/images/img-0003.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> and <img src="/MI/a85ec1d70c68f3f21ce6a17e86cb409a/images/img-0004.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" />, then this proof would leave you dead.
<h3>The law of the excluded middle</h3>
<p>Hidden within this proof is another principle Brouwer rejected, called the <em>law of the excluded middle</em>. Suppose that <em>P</em> is some statement (eg "<img src="/MI/418d667dca366d6dff8f3f52a0f62957/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is rational") and NOT <em>P</em> is its negation (eg "<img src="/MI/9d5efb924102084d5c7ac1205b803ab3/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is irrational"). In classical logic the combined statement "either <em>P</em> is true or NOT <em>P</em> is true" ("either <img src="/MI/3d6328ec7c4a136aa88987807598885a/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is rational or <img src="/MI/3d6328ec7c4a136aa88987807598885a/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is irrational") is always considered true, even if you don't know which one of the two options is true. This makes sense, if it's not one then it has to be the other, but it's deeply unsatisfying. If I told you that either I will come to see you tomorrow or I will not, you couldn't accuse me of lying, but you'd be annoyed. </p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/constructivism/clock.jpg" alt="LEJ Brouwer" width="350" height="263" />
<p>Brouwer believed that our sense of time feeds out intuition of the natural numbers.</p>
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<p>Intuitionists treat the statement "either <em>P</em> is true or NOT <em>P</em> is true" accordingly: they see it as meaningless unless you have constructed, in front of your mental eye, a proof of one of the two components. You shouldn't build mathematical proofs, or plan your personal life, according to empty statements such as "<img src="/MI/d057cc6036522c1d5cd6dcecb2896959/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is rational or <img src="/MI/d057cc6036522c1d5cd6dcecb2896959/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is irrational" or "I'll come to see you tomorrow or I won't".</p>
<p>The law of the excluded middle affords mathematicians a bread-and-butter technique they use almost every day: <em>proof by contradiction</em>, which shows that a statement <em>P</em> is true by showing that its negation NOT <em>P</em> must be false, but without constructing an explicit proof of <em>P</em>. A famous example is the proof that <img src="/MI/fe34a52857184efc7ae3bdb4f4cca715/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> is an irrational number (see <a href="/content/maths-minute-square-root-2-irrational">here</a>). The idea of rejecting the law of the excluded middle, and hence many proofs by contradiction, spooked (and still spooks) mathematicians considerably. As the famous mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Hilbert.html">David Hilbert</a> remarked in
1928, "Taking the law of the excluded middle from the mathematician would be the same, say,
as proscribing the telescope to the astronomer or to the boxer the use of his fists."
</p>
<p>To Brouwer, however, the law of the excluded middle was outdated mathematical folklore, to be considered "as a phenomenon of the history of civilisation of the same kind as the former belief in the rationality of <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" />, or in the rotation of the firmament about the Earth."
He and his followers revised the laws of logic to reflect their ideas and proceeded to develop mathematics in an intuitionist way. Today we use the term "intuitionism" to refer to Brouwer's particular brand of <em>constructivism</em>: a school of thought that also rejects the law of the excluded middle and requires all mathematical objects to be explicitly constructed.</p>
<h3>
Zero or not?</h3>
<p>Some of the consequences of the constructivist approach are startling, at least at first sight. Consider a statement we absolutely take for granted in normal maths: "Every real number <img src="/MI/5b48811b1b5942d807a8e31c414da11f/images/img-0001.png" alt="$x$" style="vertical-align:-1px;
width:9px;
height:9px" class="math gen" /> is either equal to 0 or it is not". A constructivist would only agree if given any <img src="/MI/5b48811b1b5942d807a8e31c414da11f/images/img-0001.png" alt="$x$" style="vertical-align:-1px;
width:9px;
height:9px" class="math gen" /> you can definitely decide which of the two alternatives it is. But a real number can have an infinitely long decimal expansion. Since it's humanly impossible to check such an expansion all the way to infinity the statement seems doubtful indeed. </p>
<div style="max-width: 340px; float: right; border: thin solid grey;
background: #CCC CFF; padding: 0.5em; margin-left: 1em; font-size:
75%">
<h3>A weak counterexample</h3>
<p>The <a href="/content/mathematical-mysteries-goldbach-conjecture">Goldbach conjecture</a> asserts that every even number greater than 2 can be written as the sum of two prime numbers. So far, nobody has managed to prove it, though it has turned out to be true for all even numbers people have been able to check it for.</p>
<p>Now define a real number <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> as follows. The first digit of <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />, which is the digit before the decimal point is <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0002.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" />. Then define the <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0003.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> digit after the decimal point to be <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0002.png" alt="$0$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> if the Goldbach conjecture holds for every even number less than or equal to <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0004.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />. If it fails for some even number less than or equal to <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0004.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" />, then the <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0003.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> digit of <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> after the decimal point is <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0005.png" alt="$1$" style="vertical-align:0px;
width:6px;
height:12px" class="math gen" />. </p><p>This is a perfectly valid definition of a real number, even in the constructivist sense: we are given a finite recipe for computing every one of the number’s digits. </p><p>Clearly, <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0006.png" alt="$x=0$" style="vertical-align:0px;
width:39px;
height:12px" class="math gen" /> if and only if the Goldbach conjecture holds for all even integers, that is,<img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0006.png" alt="$x=0$" style="vertical-align:0px;
width:39px;
height:12px" class="math gen" /> if and only if the Goldbach conjecture is true. Since we do not know if the Goldbach conjecture is true, we cannot at this moment say whether <img src="/MI/cf43496bc4fffb0f78185a42edc5fff8/images/img-0006.png" alt="$x=0$" style="vertical-align:0px;
width:39px;
height:12px" class="math gen" /> or not.</p></div>
<p>To bring this idea home Brouwer found a way of defining real numbers in terms of certain unsolved maths problems — problems that had puzzled the best minds for decades or even centuries. </p><p>Given such an unproven mathematical conjecture, Brouwer defined the corresponding real number so that it's zero if the conjecture is true and non-zero otherwise (see the box for an example). Since we don't yet know if the conjecture is true or false, and may not know it for a long time to come, we definitely can't say whether the real number is equal to zero or not. Thus, the general statement "every real number <img src="/MI/5b48811b1b5942d807a8e31c414da11f/images/img-0001.png" alt="$x$" style="vertical-align:-1px;
width:9px;
height:9px" class="math gen" /> is either equal to 0 or it is not", and by extension the law of the excluded middle, has "no mathematical reality" as Brouwer put it.</p>
<p>This argument is called a <em>weak counterexample</em> to the claim that every number is either 0 or not (and to the law of the excluded middle). It's a counterexample because it suggests (according to Brouwer) that the statement doesn't generally apply. But it's only a weak counterexample because it's time-dependent: as soon as someone solves the maths problem associated to the real number, we can definitely decide if the real number is 0 or not, so the argument ceases to be a counterexample. But in that case there'll still be other counterexamples, linked to different unsolved problems.</p>
<p>No law of the excluded middle, not being able to assume that a number is either 0 or not — that doesn't bode well. So does standard mathematics still stand up to the intuitionist/constructivist approach? Find out in the <a href="/content/paradise">second part of this article</a>.</p>
<hr/><h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-can-maths-exist-if-you-cant-see-it">here</a> to see more articles about constructivism. </p>
</div></div></div>Wed, 18 Jul 2018 14:17:13 +0000Marianne7037 at https://plus.maths.org/contenthttps://plus.maths.org/content/intuitionism#commentsParadise lost and rescued
https://plus.maths.org/content/paradise
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/constructive_icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In the <a href="/content/constructivism-2">first part of this article</a> we looked at intuitionist/constructivist mathematics as it was developed by LEJ Brouwer. At the time mathematicians feared that Brouwer's ideas would lead to a complete overhaul of mathematics. Were they right? </p>
<div class="rightimage" style="max-width: 241px;"><img src="/issue49/features/wilson/Brouwer.jpeg" alt="Luitzen Egbertus Jan Brouwer, 1881-1966, thinking hard about crumpled paper." width="241" height="326" />
<p>Luitzen Egbertus Jan Brouwer, 1881-1966.</p>
</div>
<p>At first it seemed that they might be: Brouwer proved results that appeared to fly in the face of standard mathematics. Many of those affected the real numbers and their rather slippery, infinite nature, which Brouwer himself worried about a lot.</p>
<h3>Paradise lost?</h3>
<p> We tend to take the real numbers for granted, but there are actually several ways of thinking about them, each subtly different from the others: as points along an infinitely long line, in terms of decimal expansions, and as limits of infinite sequences of rational numbers, to name three. Along with many others Brouwer favoured a definition in terms of infinite sequences (you don't need to understand it for the purpose of this article, but you can find out more about the sequence definition <a href="/content/maths-minute-real-numbers-and-cauchy-sequences">here</a>).</p>
<p> At first sight you might think that constructivists should shun infinite sequences: after all, it is humanly impossible to write one down. But constructivists aren't finitists. They are happy to accept infinite sequences as long as there is a finite recipe that tells you how to work out each individual entry of a sequence. Something like "the <img src="/MI/148194cd4654a9abd3262863a45eed2a/images/img-0001.png" alt="$nth$" style="vertical-align:0px;
width:25px;
height:11px" class="math gen" /> entry of the sequence is <img src="/MI/148194cd4654a9abd3262863a45eed2a/images/img-0002.png" alt="$1/n$" style="vertical-align:-4px;
width:25px;
height:16px" class="math gen" />", or even a sequence constructed from the Goldbach conjecture as in <a href="/content/constructivism-2">first part of this article</a>, is a permissible constructivist object.
</p>
<p> Brouwer was also worried, however, about that other intuition we have about the real numbers: that they should merge together to form a continuous line. This can only happen if, loosely speaking, there are "sufficiently many" real numbers. Brouwer worried that allowing only pre-determined sequences to define real numbers wouldn't provide sufficiently many, so he also allowed what are called <em>choice sequences</em> into the picture. These aren't rigorously pre-determined, but can result from an "ideal mathematician" making the sequence up as they go along, for example by rolling a die infinitely many times.</p>
<p> In classical mathematics the notion of such a sequence would be accepted in its infinite totality, but, being a constructivist, Brouwer insisted that we can only ever assume to know a finite part of it. This balancing act between allowing all kinds of sequences on the one hand, but being constructivist about them on the other, ultimately led to results that appeared to contradict accepted mathematics. If you are familiar with the terminology, one of those results was that any function of the real numbers on a closed interval is continuous, something that most emphatically isn't the case in standard maths. </p>
<p>These results didn't really conflict with accepted maths, but only represented a radically different view point — a function in Brouwer's maths wasn't quite the same as a function in accepted maths, for example. But together with Brouwer's somewhat fuzzy way of explaining himself and his apparent desire to destroy standard maths,
they didn't gain intuitionism much favour with mainstream mathematicians. </p>
<h3>Paradise rescued</h3>
<p> A shake-up of the area arrived in 1967 when <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Bishop.html">Errett Bishop</a> decided to look at constructivism in a more pragmatic way. Bishop agreed with Brouwer about the natural numbers: they are given to us by intuition, he thought, and are the "primary concern" of mathematics. But while those natural numbers might be god-given (as <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Kronecker.html">Leopold Kronecker</a>
metaphorically described them) Bishop insisted that the maths we do with them be of a squarely human nature. "If God has mathematics of his own to be done, let him do it himself," he wrote in his highly entertaining <em><a href="https://books.google.co.uk/books?id=GF8lBAAAQBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false">Constructivist manifesto</a></em>.</p> </p>
<div class="leftimage" style="width: 450px;"><img src="/issue49/features/wilson/MathsConstructivism_web.jpg" alt="Constructive mathematics" width="450" height="363" />
<p>Should all of mathematics be explicitly constructed?</p>
</div>
<p>Bishop also didn't agree with Brouwer's tendency to "metaphysical speculation", especially about the continuum that should be formed by the real numbers. "A bugaboo of both Brouwer and the logicians has been compulsive speculation about the nature of the continuum," he wrote bitingly in his manifesto. " [...] In Brouwer's case there seems to be a nagging suspicion that unless he personally intervened to prevent it, the continuum would turn out to be discrete."</p>
<a name="example"></a>
<p>Bishop's aim was to give "numerical meaning" to abstract mathematics. He accepted Brouwer's revised laws of logic and insisted that everything be done in "finite routines". Defining the real numbers in terms of pre-determined sequences of rational numbers, and not bothering with choice sequences, he was away: in his book <em><a href="https://books.google.co.uk/books/about/Foundations_of_constructive_analysis.html?id=o2mmAAAAIAAJ&redir_esc=y">Foundations of constructive analysis</a></em> Bishop reproved important results of analysis in a constructive way, or at least gave weaker versions that were good enough for practical purposes (if you know basic analysis, then see <a href="/content/examples-constructive-results-analysis">here</a> for an example).</p>
<p> "The extent to which good constructive substitutes exist for the theorems of classical mathematics can be regarded as a demonstration that classical mathematics has a substantial underpinning of constructive truth," Bishop wrote in the manifesto. In other words: constructivism hadn't killed mathematics as we know it, it simply required different methods of proof and led to slightly different results. </p>
<p>Bishop published his <em>Foundations of constructive analysis</em> just over fifty years ago. During this time various mathematicians have developed constructivists mathematics in various ways. Also during this time digital technology has taken the world by storm. It has even made inroads into abstract mathematics, the ultimate paper-and-pencil activity. Computers are constructivist machines: finite routines is what they run on. So how does constructivism fare today, both in view of the mathematical theory and in view of computer sciences and even artificial intelligence? We'll explore this question in a future article.</p>
<hr/><h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-can-maths-exist-if-you-cant-see-it">here</a> to see more articles about constructivism. </p>
<a name="talk"></a>
</div></div></div>Wed, 18 Jul 2018 13:06:47 +0000Marianne7050 at https://plus.maths.org/contenthttps://plus.maths.org/content/paradise#commentsWhat can we see?
https://plus.maths.org/content/what-can-we-see
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/eyes_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The human eye is a marvellous thing. You, I, and most people without vision problems, can see objects further away than we could travel in our lifetimes. The Sun, clearly visible on this cloudless summer's day, is 149 million kilometres away. And tonight, when I look at the stars, I'll be looking across light years. "We can see pretty much as far as you like," says <a href="http://www.damtp.cam.ac.uk/user/bca20/index.html">Ben Allanach</a>, professor of theoretical physics at the University of Cambridge. "If you look at a star, that's light years away, you can see it as long as the object is reasonably bright."
</p>
<h3>Seeing stars</h3>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2018/measurement/Airy_disk_spacing_near_Rayleigh_criterion.png" alt="Illustration of airy discs" width="250" height="383" />
<p>Two airy disks at various spacings.</p>
</div>
<!-- Image in public domain -->
<p>
There are limits, though, to what our eyes can see. One of these is <em>angular resolution</em>. Your eye sees light rays reflected off (or generated by) objects. These light rays pass through your pupil and are focussed by the lens in your eye on the retina. The size of your pupil (similar to the <em>aperture</em> of a camera) determines by how much you can tell two things apart depending on the angle between the corresponding light rays from those objects.
</p>
<p>
Light consists of waves and, just as a flow of water will spread out after it has passed under a narrow bridge, the light spreads out – or <em>diffracts</em> – as it passes through a gap. This can lead to light waves interfering and creating diffraction patterns as has been famously demonstrated by the <a href="/content/physics-minute-double-slit-experiment-0">double slit experiment</a>. This spreading of light creates an <em>airy disc</em> – the limit to how focussed a spot of light can be after the light has passed through a circular hole. Two different light sources can only be distinguished if their airy discs do not overlap. This limit, the <em>angular resolution</em>, is described by the angle <img src="/MI/0fb7a815c894e6b9e3a76fea16748d89/images/img-0001.png" alt="$\theta $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />, between the light rays as they enter the eye. The angular resolution depends on the size of the hole (<img src="/MI/0fb7a815c894e6b9e3a76fea16748d89/images/img-0002.png" alt="$d$" style="vertical-align:0px;
width:9px;
height:11px" class="math gen" />, its diameter measured in metres) and the wavelength (<img src="/MI/0fb7a815c894e6b9e3a76fea16748d89/images/img-0003.png" alt="$\lambda $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />) of the light: <table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/0fb7a815c894e6b9e3a76fea16748d89/images/img-0004.png" alt="\[ \theta \approx 1.22 \frac{\lambda }{d}. \]" style="width:77px;
height:33px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> The human eye has an angular resolution of about 1 arcminute (0.02 degrees or 0.0003 radians) which enables us to distinguish things that are 30 centimetres apart at a distance of 1 kilometre. "One of the stars that you see might actually be two stars that are separated by a really tiny angle," says Allanach. "You wouldn't be able to resolve them, they would just look like one star."
</p>
<h3>Over the rainbow</h3>
<p>
Another limit to our vision is related to the frequency of the light we can see. Light consists of electromagnetic waves travelling at approximately 3x10<sup>8</sup> metres per second. These waves can come with different frequencies and corresponding wavelengths. The colours we see result from the wavelength and frequency of the light wave – red has a wavelength of 620-750 nanometres and a frequency of 400-484 terrahertz, across to violet at 280-450 nanometres and 668-789 terrahertz.
</p>
<div class="leftimage" style="max-width: 300px;"><img src="/content/sites/plus.maths.org/files/articles/2011/rainbow/spectrum.jpg" height="57" width="300" alt="Spectrum" /><p >The visible spectrum from violet (left) to red (right).</p></div>
<!-- Image from Wikipedia in public domain -->
<p>
The light we can see spans the colours of the rainbow, but light waves fill the air around us in frequencies above and below that visible spectrum. Eyes of different construction to ours can see this ultraviolet or infrared light. "Humans don't see ultraviolet, but insects do," says Allanach. For example, as well as having compound eyes made up of thousands of lenses, bees also have additional light receptor cells that respond to ultraviolet light. (You can read more about the super powers of bees <a href="https://www.beeculture.com/bees-see-matters/">here</a>.)
</p><p>
An interesting illustration of observation of colour comes from the Norwegian Neil Harbisson. Harbisson is completely colour-blind, a very rare condition that only happens in men. "He sees the world in black and white," says Allanach. But Harbisson, who recognises himself as a cyborg, has enhanced his body to allow him to experience colour. He's had a camera implanted on his head that turns colours it captures into sound.
</p><p>
"The sound gets transmitted through his bone, into his ear," says Allanach. "He can hear colours. When the phone rings, it sounds purple." Harbisson's experience of colour switches one sense – sight – for another – hearing – something called <em>synesthesia</em>. "And he's had it enhanced so he can see, for example, infrared light. Apparently when someone uses an old-school remote near him, he hears it." (You can watch a TED Talk by Harbisson <a href="https://www.ted.com/talks/neil_harbisson_i_listen_to_color">here</a>.)
</p>
<h3>More than human</h3>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/measurement/Neil_Harbisson_cyborgist.jpg" alt="Neil Harbisson" width="350" height="233" />
<p>Neil Harbisson. Image: <a href="https://commons.wikimedia.org/wiki/File:Neil_Harbisson_cyborgist.jpg"> Dan Wilton/The Red Bulletin</a>, <a href="https://creativecommons.org/licenses/by/2.0/deed.en">CC BY 2.0</a>.</p>
</div>
<p>
This sounds extreme, but we've been in the business of enhancing our bodies and our senses for a long time. From ear trumpets, to glasses and contact lenses, and the advancements of cochlear implants – humans have been finding ways to overcome their physical limitations. And scientifically we have been extending our observational abilities since the seventeenth century. The ancient Greeks used magnifying glasses and the development of microscopes in the seventeenth century allowed new insights in biology. <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Galileo.html">Galileo Galilei</a> built one of the first telescopes and discovered moons orbiting Jupiter in 1610. As bigger lenses are used you increase the angular resolution, allowing astronomers to see smaller objects and separate stars that are really close.
<p>
Microscope and telescopes both use lenses to blow up the magnification of the observed objects. "But when you get down to a certain level, say of a molecule, no [optical] microscope is going to help you see that," says Allanach. So you have to use other devices such as <a href="https://en.wikipedia.org/wiki/Electron_microscope">electron microscopes</a> and <a href="https://en.wikipedia.org/wiki/Scanning_probe_microscopy">scanning probe microscopes</a>.
</p><p>
Mathematics and technology can now allow us to see so much more, even inside our very selves. <em>Magnetic resonance imaging</em> (MRI) works by making the magnetic fields in water molecules oscillate and detecting the resulting disturbances of an electromagnetic field. (You can read more about MRI <A href="https://plus.maths.org/content/uncovering-mathematics-information">here</a>.) MRI machines reconstruct an image using mathematical techniques implemented in a computer program, allowing us to see where the water is in the body. Rather than just amplifying our human senses, like glasses or a microscope, the machine reconstructs the image for us from the information it senses. "It's not seeing in the usual sense," says Allanach.
</p><p>
But is it really that different from our normal human experience of seeing? "When we see something, nothing gets to the centre of our brain without going through some visual processing," says Allanach. "There's already processing before you're aware of it, before your CPU, the computer in the middle of your brain, gets the information. So it's not as different observing something through some instrument that you've built, than it is just observing it yourself."
</p>
<hr><h3>About the article</h3>
<div class="leftimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2015/LHC/metalksmall.jpg" alt="" width="200" height="239" />
</div>
<p><a href="http://users.hepforge.org/~allanach/">Ben Allanach</a> is a Professor in Theoretical Physics at the <a href="http://www.damtp.cam.ac.uk/">Department of Theoretical Physics and Applied Mathematics</a> at the University of Cambridge. His research focuses on discriminating different models of particle physics using LHC data. He worked at CERN as a research fellow and continues to visit frequently.</p>
<p><a href="/content/people/index.html#rachel">Rachel Thomas</a> is Editor of <em>Plus</em>. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-limits-observation">here</a> to see more articles about the limits of observation. </p>
</div></div></div>Wed, 11 Jul 2018 22:04:05 +0000Rachel7047 at https://plus.maths.org/contenthttps://plus.maths.org/content/what-can-we-see#commentsWhat can we agree to look for?
https://plus.maths.org/content/what-can-we-agree-look
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/LISA_icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In 2012 science made the headlines with the <a href="/content/higgs">discovery of the Higgs boson</a>. The Higgs boson was the missing link in the standard model of particle physics, a mathematical description of the fundamental particles and their interactions. Although phenomenally successful in most of its predictions, the standard model could not explain why some of the fundamental particles have mass. The physicist <a href="https://en.wikipedia.org/wiki/Peter_Higgs">Peter Higgs</a> came up with a possible explanation in the 1960s, which predicts the existence of the particle that now carries his name. </p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/measurement/LISA.jpg" alt="LISA" width="350" height="197" />
<p>An artist's rendering of the <a href="https://lisa.nasa.gov/">Laser Interferometer Space Antenna</a> (LISA), another huge scientific project which will be launched in 2030. Image: Credits: ESA/C. Carreau.</p>
</div>
<p>But discovering the Higgs boson took more than fifty years, thousands of people, huge sums of money and collaboration from countries around the world. The discovery was only possible by building the largest scientific experiment ever made – the Large Hadron Collider (LHC), a 27km long ring-shaped tunnel in which protons are made to collide at extremely high energies. These particle collisions produce Higgs bosons, and when it was finally observed in 2012 physicists had seen everything in the standard model. (You can read more about the discovery of the Higgs boson in <a href="/content/what-can-science-see"><em>What can science see</em></a> and in <a href="/content/higgs"><em>The Higgs boson: a massive discovery</em></a>, and about the LHC in <a href="/content/particle-hunting-lhc"><em>Particle hunting at the LHC</em></a>.)</p>
<p>There are already some contenders, in theory. "There's this theory called supersymmetry, which I've worked on a lot," says <a href="http://www.damtp.cam.ac.uk/user/bca20/index.html">Ben Allanach</a>, professor of theoretical physics at the University of Cambridge. "It predicts lots of new particles and it explains some mysteries about the Higgs boson that we don't currently understand. It seems like a really beautiful theory. When you really understand it, you feel in your heart of hearts that this has to be right." A lot of people agree with Allanach and many were expecting to see evidence for supersymmetry from the latest run of the LHC. "But the LHC hasn't seen these new particles yet. I'm kind of still expecting them to show up but they're late to the party, it's got to be said."</p>
<p>The problem might be that the supersymmetric particles are just too heavy for the LHC to produce. "You're always turning energy into mass at the LHC and vice versa. The LHC only has a certain amount of energy in it - you could only produce something that's about 6,500 times the mass of the proton." If the particle you want to observe is heavier than this limit, there is no way it would be observed by the LHC. </p>
<p>For the case of supersymmetry, the more you run the LHC and don't see the predicted particles, the heavier the particles must be to have been missed. As this happens people's view on the usefulness and beauty of the theory begin to change. "The thing is, the heavier you make these particles [in your theory], the less you solve the mysteries," says Allanach. "Taste [also] plays a role. At some point they're too heavy, and they don't solve this aesthetic problem that we have with the Higgs, and you give up. Many people have. Some of the rest of us think that's premature."</p>
<h3>Economic and political limits</h3>
<p>The obvious approach to this problem is to build a new, bigger collider with even more energy. But the limits to that aren't just technological and scientific. "It costs billions to run one of these big experiments. It involves tens of thousands of people. But for the foreseeable future, unless there's some major technological breakthrough where you can get huge energies out of some other technology, it looks like that's going to continue."</p>
<p>The problem is not just financial, it's also political. It takes the collaboration between scientific communities and governments of many countries to fund and build such projects. "You really need a worldwide collaboration. This is ulta-specialised, cutting-edge science, and you need the best scientists from around the world to be able to push this forward."</p>
<p>"That's quite a politically difficult thing to do, especially if you don't know what you're going to find." When the LHC was conceived and set up in the 1980s not everyone believed in the Higgs boson. "But we had this no-lose theorem: we'll find the Higgs boson, or something like it, with this machine. At the moment I'm working on a no-lose theorem for the next collider."</p>
<p>We've seen brilliant discoveries in the last decade – the Higgs boson at the LHC, <a href="/content/maths-minute-gravitational-waves">gravititational waves</a> at LIGO – thanks to huge collaborative experiments. New, bigger experiments are in the pipeline. The <a href="https://lisa.nasa.gov/">Laser Interferometer Space Antenna</a> (LISA) is in development and set to launch in 2030. Allanach and his colleagues are building the case for a new, larger collider. Humanity, through science, is always pushing the limits of what it can observe. And what these new enormous experiments will allow us to see, we can't wait to find out! </p>
<hr><h3>About the article</h3>
<div class="leftimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2015/LHC/metalksmall.jpg" alt="" width="200" height="239" />
</div>
<p><a href="http://users.hepforge.org/~allanach/">Ben Allanach</a> is a Professor in Theoretical Physics at the <a href="http://www.damtp.cam.ac.uk/">Department of Theoretical Physics and Applied Mathematics</a> at the University of Cambridge. His research focuses on discriminating different models of particle physics using LHC data. He worked at CERN as a research fellow and continues to visit frequently.</p>
<p><a href="/content/people/index.html#rachel">Rachel Thomas</a> is Editor of <em>Plus</em>. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-limits-observation">here</a> to see more articles about the limits of observation.</p>
</div></div></div>Wed, 11 Jul 2018 13:56:47 +0000Marianne7049 at https://plus.maths.org/contenthttps://plus.maths.org/content/what-can-we-agree-look#commentsWhat can science see?
https://plus.maths.org/content/what-can-science-see
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_30.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>In a <a href="/content/what-can-we-see">previous article</a> we looked at the some of the limits of human observation. In this article we explore some of the limits to scientific observation.</em></p>
<div class="rightimage"><img src="/content/sites/plus.maths.org/files/news/2012/higgs/atlashiggs4electrons_web.jpg" width="350" height="311" alt="ATLAS decay to 4 electrons"/><p style="width:350px;">An event recorded by the ATLAS detector in 2012 showing the characteristics expected from a Higgs boson decaying into four electrons. (Image <a href="http://www.atlas.ch/news/2012/latest-results-from-higgs-search.html">ATLAS</a>)</p></div>
<p>CERN's <a href="https://home.cern/topics/large-hadron-collider">Large Hadron Collider</a> (LHC) is the world’s largest, and most famous, laboratory experiment. It's a 27 kilometre circular tunnel that sits 100m below ground on the Franco-Swiss border. Since it was switched on in 2008 the LHC has accelerated tiny particles of matter to close to the speed of light, smashing them together in over <a href="http://cds.cern.ch/record/2059631/files/CERN-Brochure-2015-003-Eng.pdf">a million billion collisions</a>.
</p>
<p>"At the Large Hadron Collider at CERN, we built what's essentially an enormous microscope," says <a href="http://www.damtp.cam.ac.uk/user/bca20/index.html">Ben Allanach</a>, professor of theoretical physics at the University of Cambridge. "The bigger you build the thing, the smaller [scales] you can see." It might not look like a microscope, but that is because it is using a different way to see the very small. </p>
<p>At the LHC they are not using light to detect particles. Instead this giant microscope fires protons around a circular tunnel at very high energies and then captures the collisions using machines that act like 3D cameras. It's in these collisions that new particles emerge. "Then you take all the data, the electronic impulses [from the cameras], and you try to do some detective work to work out what was happening." (You can find out more in Allanach's article <a href="/content/particle-hunting-lhc"><em>Particle hunting at the LHC</em></a>.)
</p>
<h3>Discovering the Higgs</h3>
<div class="leftimage" style="width:287px" width="287" height="317"><img src="/content/sites/plus.maths.org/files/news/2011/higgs/higgs_web.jpg" alt="Histogram of the number of events that produced photons with a particular energy"/><p>A histogram showing a simulation of the number of collisions at the LHC that would produce
pairs of photons with a particular energy, for a Higgs boson of mass 120 GeV. The dotted line is the predicted counts of collisions that do not involve the Higgs (as predicted by the Standard Model). A significant bump in the data deviating from this shape can be explained by collisions that produced a Higgs boson, which subsequently decayed to
photons. Image: Ben Allanach.</p></div>
<p>
The LHC hit the headlines in 2012 with its discovery of the <em>Higgs boson</em>. Until then, the Higgs had been a missing link in the standard model of particle physics, a mathematical description of the fundamental particles and their interactions. But finding the Higgs wasn't a matter of directly catching the beast itself. </p>
<p>"The Higgs leaves its signature in the properties of the particles it decays into," says Allanach. The Higgs boson was detected by looking for collisions in which it decayed into two particles of light. Only about one in five hundred Higgs particles decays in this way, the rest decay in a way that is more difficult to see. "You do lots and lots of collisions, you filter them and look for the ones that only have two particles of light,"says Allanach. These photons, these particles of light, have quite a lot of energy. "If there's a Higgs boson, the energies of these photons should add up to the mass of the Higgs boson."</p>
<p>There are ordinary processes which result in collisions that do not produce a Higgs boson, which also give two photons. The probability of these other collisions occurring varies smoothly with the energy of the photons produced: the high-energy photons aren't produced often, but the low-energy ones are produced a lot. "It's a smooth distribution," says Allanach. </p>
<p>
When the beams of protons collide in the LHC, detectors measure the energy, momentum and direction of the particles produced. The results are plotted in histograms counting how many collisions produced certain footprints, say pairs of photons with given energies as that shown in the histograms below. "But the Higgs bosons all come out with one particular energy, so what you're looking for is a smooth distribution with a big spike in it. That spike is your Higgs boson." (See the <em>Plus</em> article <em><a href="/content/countdown-higgs">Countdown to the Higgs</a></em> for more detail.)</p>
<h3>How sure?</h3>
<p>The collisions inside the LHC are quantum processes, so inherently, by the laws of quantum physics, they are random processes. "You can't say on a particular collision what the energy [of the two photons] is going to be, or even if it's going to give you two photons, but you can say: if we've got the theory right we know what the relative probabilities of the different possibilities are," says Allanach. "We made some predictions with the mathematics of the theory, as to how the collisions should look like if the Higgs boson was there, and how they should look if it's not there."</p>
<div class="rightimage"><img src="/content/sites/plus.maths.org/files/news/2012/higgs/cms2photons_web.jpg" width="300" height="289" alt="Data from CMS for decays into two photons"/><p style="width:300px;">The data from the CMS detector for decays into two photons. The bump at 125-126GeV indicates the presence of the new particle. (Image <A href="http://cms.web.cern.ch/news/observation-new-particle-mass-125-gev">CMS</a>) </p></div>
<p>When data first starts coming in from the LHC it looks quite noisy, but the more data you collect the more the histogram starts to even out. When peaks appear in the data physicist need to ask how likely it is that that the peak is due to something — like the Higgs boson being produced — rather than just a random fluctuation from the fuzziness of quantum physics.</p>
<p>
"There are two thresholds we use in particle physics," says Allanach. "One of them is evidence for an effect: which means it's interesting but we're not sure. That happens if the probability of this peak is one in about a thousand." Such a probably means that if we were to run a thousand Large Hadron Collider experiments with the same conditions, creating a thousand full runs of data, then only one of them would register a peak that big just by chance, without there actually being a Higgs boson. That sounds pretty good evidence but this isn't good enough for particle physics.</p><p>
"The gold standard for observation — for discovery — is about one in 10 million. There's a peak [in the histogram such] that only one in ten million [repeated LHC experiments] would you get such a big peak by chance." Then, and only then, are particle physicists happy to say that what they've seen is not due to chance, and they have observed, for the first time, a new particle.</p>
</p>
<hr><h3>About the article</h3>
<div class="leftimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2015/LHC/metalksmall.jpg" alt="" width="200" height="239" />
</div>
<p><a href="http://users.hepforge.org/~allanach/">Ben Allanach</a> is a Professor in Theoretical Physics at the <a href="http://www.damtp.cam.ac.uk/">Department of Theoretical Physics and Applied Mathematics</a> at the University of Cambridge. His research focuses on discriminating different models of particle physics using LHC data. He worked at CERN as a research fellow and continues to visit frequently.</p>
<p><a href="/content/people/index.html#rachel">Rachel Thomas</a> is Editor of <em>Plus</em>. </p>
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<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-limits-observation">here</a> to see more articles about the limits of observation.</p>
</div></div></div>Wed, 11 Jul 2018 11:48:48 +0000Marianne7048 at https://plus.maths.org/contenthttps://plus.maths.org/content/what-can-science-see#commentsWho's watching: The limits of observation
https://plus.maths.org/content/whos-watching-limits-observation
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_6_0.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Physics is all about observation, but how much can we actually see? These articles explore some of the limits of observation — be they natural, scientific, political, or down to the jelly-like quantum nature of reality. </p>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/eyes_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/what-can-we-see">What can we see?</a> — The human eye is a marvellous thing: it can see pretty much as far as you like. But what are its limits when it comes to distinguishing objects and seeing colours? </p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_30.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/what-can-science-see">What can science see?</a> — Observing the smallest building blocks of nature — such as the famous Higgs boson — isn't about seeing in the ordinary sense. It's about careful mathematical detective work and statistical analysis.</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/LISA_icon_0.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/what-can-we-agree-look">What can we agree to look for?</a> — Even if the science and technology you need to observe something exist, there can still be political and economical limits to what can be done.</p></div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_6.png" alt="" width="100" height="100" /> </div><p><a href="/content/heisenbergs-uncertainty-principle">Heisenberg's uncertainty principle</a> — One of the most famous results from quantum mechanics puts limit on the accuracy with which we can observe fundamental particles. It's like squeezing jelly! </p></div>
<br class="brclear"/>
<hr/>
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<p><em>This package is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-physics-observers">here</a> to see more articles and videos about questions to do with observers in physics. </em></p></div></div></div>Wed, 11 Jul 2018 09:28:14 +0000Marianne7052 at https://plus.maths.org/contenthttps://plus.maths.org/content/whos-watching-limits-observation#commentsMaths in a minute: Truth tables
https://plus.maths.org/content/maths-minute-truth-tables
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/light_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In standard mathematical logic every statement — "the cat is white", "the dog is black", "I am hungry" — is considered to be either true or false. Given two statements <em>P</em> and <em>Q</em>, you can make more complicated statements using logical connectives such as AND and OR. </p>
<p>For example, the statement <em>P</em> AND <em>Q</em> (eg "the cat is white and the dog is black") is only considered true if both <em>P</em> and <em>Q</em> are true, otherwise it is false. This can be summarised in a truth table:</p>
<table class="data-table">
<tr><td><em>P</em></td><td><em>Q</em></td><td><em>P</em> AND <em>Q</em></td></tr>
<tr><td>T</td><td>T</td><td>T</td></tr>
<tr><td>T</td><td>F</td><td>F</td></tr>
<tr><td>F</td><td>T</td><td>F</td></tr>
<tr><td>F</td><td>F</td><td>F</td></tr>
</table>
<p>The table lists every combination of truth values for <em>P</em> and <em>Q</em> and then tells you what the corresponding truth value for <em>P</em> AND <em>Q</em> is.</p>
<p>Similarly, the OR connective is defined by the following table:</p>
<table class="data-table">
<tr><td><em>P</em></td><td><em>Q</em></td><td><em>P</em> OR <em>Q</em></td></tr>
<tr><td>T</td><td>T</td><td>T</td></tr>
<tr><td>T</td><td>F</td><td>T</td></tr>
<tr><td>F</td><td>T</td><td>T</td></tr>
<tr><td>F</td><td>F</td><td>F</td></tr>
</table>
<p>There is also a truth table that defines NOT <em>P</em>, the negation of a statement <em>P</em> (if <em>P</em> is "the cat is white" then NOT <em>P</em> is "the cat is not white"). Unsurprisingly, NOT <em>P</em> is true when <em>P</em> is false and vice versa:</p>
<table class="data-table">
<tr><td><em>P</em></td><td>NOT <em>P</em></td></tr>
<tr><td>T</td><td>F</td></tr>
<tr><td>F</td><td>T</td></tr>
</table>
<p>Using the OR and the NOT operators, we can derive the <em>law of the excluded middle</em>, which says that <em>P</em> OR NOT <em>P</em> is always true:</p>
<table class="data-table">
<tr><td><em>P</em></td><td>NOT <em>P</em></td><td><em>P</em> OR NOT <em>P</em></td></tr>
<tr><td>T</td><td>F</td><td>T</td></tr>
<tr><td>F</td><td>T</td><td>T</td></tr>
</table>
<p>Using truth tables you can figure out how the truth values of more complex statements, such as</p>
<p> <em>P</em> AND (<em>Q</em> OR NOT </em>R</em>)</p>
<p> depend on the truth values of its components. We have filled in part of the truth table for our example below, and leave it up to you to fill in the rest.</p>
<table class="data-table">
<tr><td><em>P</em></td><td><em>Q</em></td><td><em>R</em></td><td>NOT <em>R</em></td><td><em>Q</em> OR NOT </em>R</em></td> <td><em>P</em> AND (<em>Q</em> OR NOT </em>R</em>)</td></tr>
<tr><td>T</td><td>T</td><td>T</td><td>F</td><td>T</td><td></td></tr>
<tr><td>T</td><td>T</td><td>F</td><td>T</td><td>T</td><td></td></tr>
<tr><td>T</td><td>F</td><td>T</td><td>F</td><td>F</td><td></td></tr>
<tr><td>F</td><td>T</td><td>T</td><td>F</td><td>T</td><td></td></tr>
<tr><td>T</td><td>F</td><td>F</td><td>T</td><td>T</td><td></td></tr>
<tr><td>F</td><td>T</td><td>F</td><td>T</td><td>T</td><td></td></tr>
<tr><td>F</td><td>F</td><td>T</td><td>F</td><td>F</td><td></td></tr>
<tr><td>F</td><td>F</td><td>F</td><td>T</td><td>T</td><td></td></tr>
</table>
<p>If you have enjoyed doing this, you could also define your own logical connectives using truth tables. Or you could read a text book on Boolean logic or propositional logic.</p>
<hr/><h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-can-maths-exist-if-you-cant-see-it">here</a> to see more articles about constructivism. </p> </div></div></div>Wed, 04 Jul 2018 16:14:38 +0000Marianne7046 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-truth-tables#commentsSomething from nothing?
https://plus.maths.org/content/something-nothing
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/rabbit_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>"You can take it or leave it" is a phrase we use purely for effect. It's obvious that the person we're talking to is either going to take it or leave it — what else can they do? — so in terms of information content, the phrase is useless.
It's an example of a
logical
<em>tautology</em>: something that is always true, no matter the circumstance. That's probably why the phrase is so annoying.</p>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2018/LEM/rabbit.jpg" alt="Rabbit in hat" width="250" height="353" />
<p>Proving things non-cnostructively is a bit like pulling a rabbit out of a hat.</p>
</div>
<p>Curiously, mathematicians frequently draw meaning from such vacuous tautologies. The phrase above is of the form <em>P</em> OR NOT <em>P</em>, where <em>P</em> is some statement ("you can leave it") and NOT <em>P</em> is its negation ("you can not take it"="you can leave it"). According to the classical rules of logic, <a href="/content/maths-minute-truth-tables">such a statement is always true</a> — put differently, if <em>P</em> is true then NOT <em>P</em> is false and vice versa.</p>
<p> This so-called <em>law of the excluded middle</em>
allows mathematicians to perform logical magic. Suppose you want to show that
some statement <em>P</em> is true. If you can show that NOT
<em>P</em> is definitely false, then <em>P</em> has no choice but being true and you're done. You have proved
the statement <em>P</em> without necessarily getting your hands dirty with the
details of <em>P</em>. The technique is called a <em>proof by contradiction</em>, or <em>reductio ad absurdum</em> to use the Latin description.</p>
<p>If this sounds confusing, here is an example. A prime number is a
natural
number that is only divisible by itself and 1. The first few primes
are 2, 3, 5, and 7. The primes are like building blocks for all the
natural numbers, because every natural number that is not itself a
prime can be written as a product of primes in a unique way. For
example, 24 = 2x2x2x3.</p>
<p>We are going to prove that there are infinitely many prime numbers using a proof by contradiction. Suppose there are only finitely many and label them <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0001.png" alt="$p_1$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0002.png" alt="$p_2$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0003.png" alt="$p_3$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, up to <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0004.png" alt="$p_ n$" style="vertical-align:-3px;
width:17px;
height:10px" class="math gen" />. Now consider the number </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0005.png" alt="\[ p=p_1p_2...p_ n+1. \]" style="width:126px;
height:15px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Because it’s different from all the <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0001.png" alt="$p_1$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" /> up to <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0004.png" alt="$p_ n$" style="vertical-align:-3px;
width:17px;
height:10px" class="math gen" />, and these are all the primes there are, it cannot itself be a prime. But then, as we have noted above, it must be a product of primes. This means that one of our <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0001.png" alt="$p_1$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" /> up to <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0004.png" alt="$p_ n$" style="vertical-align:-3px;
width:17px;
height:10px" class="math gen" /> must divide it. But since the <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0001.png" alt="$p_1$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" /> up to <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0004.png" alt="$p_ n$" style="vertical-align:-3px;
width:17px;
height:10px" class="math gen" /> divide <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0006.png" alt="$p-1=p_1p_2...p_ n$" style="vertical-align:-3px;
width:122px;
height:15px" class="math gen" />, they can’t possibly divide <img src="/MI/fe33f8cccdc531c30afba561c9197614/images/img-0007.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> as well. This is a contradiction. Hence, the statement that there are only finitely many primes is false. Therefore, its negation, that there are infinitely many primes must be true. Here the statement <em>P</em> is "there are infinitely many primes" and NOT <em>P</em> is "there are finitely many primes". </p>
<p>Depending on your preference, you might find this an
elegant argument, or you might feel a little cheated. OK, so there are
infinitely many primes, but what are they? How do you find them? Our
proof is an example of a <em>non-constructive proof</em>, which shows that
something exists without actually constructing it.
</p>
<p>Mathematics, as it is accepted by most people, relies heavily on
non-constructive proofs and proofs by contradiction. As the famous
mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Hilbert.html">David Hilbert</a> remarked in
1928, "Taking the law of the excluded middle from the mathematician would be the same, say,
as proscribing the telescope to the astronomer or to the boxer the use of his fists."
</p>
<div class="leftimage" style="width: 450px;"><img src="/issue49/features/wilson/MathsConstructivism_web.jpg" alt="Constructive mathematics" width="450" height="363" />
<p>Should all of mathematics be explicitly constructed?</p>
</div>
<p>Some people believe, however, that logically such proofs live on thin ice. The proofs purport to show that something exists (or is true) by showing that it can't possibly not exist (or be false). Is that really as convincing as proving that it exists by actually constructing it? A school of thought called
<em>constructivism</em> believes that it isn't, and that we should only deal with
mathematical objects that we can get our heads around in a
concrete way. There should be a recipe, an algorithm, for calculating all the
properties of the object in question. To borrow a phrase from the
US mathematician <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Bishop.html">Errett Bishop</a>, everything in mathematics
should have "numerical meaning".</p>
<p>Constructivists reject the law of the excluded middle. To them, a statement of the form <em>P</em> OR NOT <em>P</em> is only
true when we know for certain
that one of <em>P</em> or NOT <em>P</em> is true. Until we have
proved, constructively, that there are infinitely many primes or that
there are finitely many primes, the statement "either there are
infinitely many primes or finitely many primes" means nothing, and we
certainly can't base a proof on it. </p>
<p>What does this constructivist stance mean for mathematics as whole? Our prime number proof from above can quite easily be saved: there’s a version that isn’t based on contradiction. Given any finite list of prime numbers label them <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0001.png" alt="$p_1$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0002.png" alt="$p_2$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0003.png" alt="$p_3$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, up to <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0004.png" alt="$p_ n$" style="vertical-align:-3px;
width:17px;
height:10px" class="math gen" />. Form the number </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0005.png" alt="\[ p=p_1p_2...p_ n+1 \]" style="width:122px;
height:15px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> and find all its prime factors (if <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0006.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> is itself a prime, then it’s the only number in its prime factorisation). The primes <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0001.png" alt="$p_1$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0002.png" alt="$p_2$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0003.png" alt="$p_3$" style="vertical-align:-3px;
width:15px;
height:10px" class="math gen" />, up to <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0004.png" alt="$p_ n$" style="vertical-align:-3px;
width:17px;
height:10px" class="math gen" /> cannot appear in the prime factorisation of <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0006.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> because they divide <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0007.png" alt="$p-1$" style="vertical-align:-3px;
width:36px;
height:15px" class="math gen" /> and therefore don’t divide <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0008.png" alt="$p.$" style="vertical-align:-3px;
width:13px;
height:10px" class="math gen" /> Thus, the prime factorisation of <img src="/MI/57d597f1a36573b6863463e805993dcc/images/img-0006.png" alt="$p$" style="vertical-align:-3px;
width:10px;
height:10px" class="math gen" /> provides us with at least one prime not on the original list. You can now repeat the process including the new prime(s) in your list to find yet more new prime(s). And so on. In this way, you can generate infinitely many primes. </p>
<p>Can all proofs in mathematics be rewritten in a constructive way so easily? We'll examine this question, as well as the ins and outs of constructivism, in an upcoming article. </p>
<hr/><h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. </p>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-can-maths-exist-if-you-cant-see-it">here</a> to see more articles about constructivism. </p> </div></div></div>Wed, 04 Jul 2018 11:03:19 +0000Marianne7045 at https://plus.maths.org/contenthttps://plus.maths.org/content/something-nothing#commentsHeisenberg's uncertainty principle
https://plus.maths.org/content/heisenbergs-uncertainty-principle
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_6.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The world we live in appears to be definite. Something exists or it doesn't. An object is either here or there. You are either alive or dead. Quantum physics, however, is famously counterintuitive in this sense. The quantum world is fuzzy, where the definite is replaced with probabilities; a particle can be here, or there, or a mixture of the two, and we can only predict its location with probabilities given by something called the <em>wave function</em>. (You can read more about the the role of probabilities and the wave function in <em><a href="/content/ridiculously-brief-introduction-quantum-mechanics">A ridiculously short introduction to some very basic quantum mechanics</a></em>.)
</p>
<div class="rightimage"><img src="/content/sites/plus.maths.org/files/articles/2011/conway/heisenberg_web.jpg" width="151" height="240" alt="Werner Heisenberg in 1933" /><p style="width:151px">Werner Heisenberg in 1933 (photo: <a href="http://www.bild.bundesarchiv.de">German Federal Archive</a>)</p></div>
<!-- Image found at http://en.wikipedia.org/wiki/File:Bundesarchiv_Bild183-R57262,_Werner_Heisenberg.jpg -->
<p>
This quantum fuzziness lies behind one of the most famous principles of quantum physics: <em>Heisenberg's uncertainty principle</em>. In 1927 the German physicist, <a href="https://en.wikipedia.org/wiki/Werner_Heisenberg">Werner Heisenberg</a>, framed the principle in terms of measuring the position and momentum of a quantum particle, say of an electron. (The momentum of an object is its mass times its velocity.) The uncertainty principle states that you cannot know, with absolute certainty, both the position and momentum of an electron – the more accurately you measure one of these properties the less accurately your knowledge of the other.
</p><p> If we write the uncertainty in a particle's position as <img src="/MI/f49a27295fb8db529f17ef8b504ace13/images/img-0001.png" alt="$\delta x$" style="vertical-align:0px;
width:17px;
height:11px" class="math gen" />, and the uncertainty in a particle's momentum as <img src="/MI/f49a27295fb8db529f17ef8b504ace13/images/img-0002.png" alt="$\delta \rho $" style="vertical-align:-3px;
width:17px;
height:14px" class="math gen" />, then the uncertainty of these two properties is connected in the following way: <table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/f49a27295fb8db529f17ef8b504ace13/images/img-0003.png" alt="\[ \delta x \times \delta \rho \geq \frac{\hbar }{2} \]" style="width:88px;
height:34px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> where <img src="/MI/f49a27295fb8db529f17ef8b504ace13/images/img-0004.png" alt="$\hbar $" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> is the <em><a href="https://en.wikipedia.org/wiki/Planck_constant">Planck's constant</a></em> (a kind of order of magnitude of the quantum world) divided by <img src="/MI/17b55d49a04c1c9ecbac5b97c94eb994/images/img-0001.png" alt="$2\pi $" style="vertical-align:0px;
width:18px;
height:12px" class="math gen" />. </p><p>
This is the most common mathematical expression of Heisenberg's uncertainty principle, that the product of these uncertainties has a minimum value. But the principle extends to other pairs of particles (called <em>complementary variables</em>), such as length of time and energy, whose connection is expressed in similar inequalities.
</p>
<h3>A misleading thought experiment</h3>
<p>
Heisenberg himself tried to understand this principle physically using the following thought experiment. Suppose you wanted to locate the position of an electron. To do this you would use a microscope which bounces photons of light of an object in order to observe it. The accuracy of a microscope is limited by the wavelength of the light used – the shorter the wavelength the more accurate your observations. Heisenberg suggested using a gamma-ray microscope as gamma-rays are a type of light with a very short wavelength.
</p><p> But the tradeoff is that light waves with a shorter wavelength have a higher frequency, and the photons have correspondingly higher energy. (This is because for waves, <img src="/MI/51d95ff7927f20d9377e65a9994e3fa5/images/img-0001.png" alt="$v=\lambda f$" style="vertical-align:-3px;
width:49px;
height:14px" class="math gen" /> where <img src="/MI/51d95ff7927f20d9377e65a9994e3fa5/images/img-0002.png" alt="$v$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> is the speed of the wave, <img src="/MI/51d95ff7927f20d9377e65a9994e3fa5/images/img-0003.png" alt="$\lambda $" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> is the wavelength and <img src="/MI/51d95ff7927f20d9377e65a9994e3fa5/images/img-0004.png" alt="$f$" style="vertical-align:-3px;
width:8px;
height:14px" class="math gen" /> is the frequency. The speed of light is fixed at <img src="/MI/51d95ff7927f20d9377e65a9994e3fa5/images/img-0005.png" alt="$c\approx 3\times 10^8 ms^{-1}$" style="vertical-align:0px;
width:121px;
height:14px" class="math gen" /> so as the wavelength decreases the frequency must increase. And the energy, <img src="/MI/51d95ff7927f20d9377e65a9994e3fa5/images/img-0006.png" alt="$E$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" />, of a photon is given by<img src="/MI/51d95ff7927f20d9377e65a9994e3fa5/images/img-0007.png" alt="$E=hf$" style="vertical-align:-3px;
width:54px;
height:14px" class="math gen" />.) Heisenberg reasoned you could very accurately measure an electron's position with a gamma-ray microscope, but to do so you have to bounce at least one gamma-ray photon off the electron. But the energy of gamma-ray photons is so high, the collision would affect the motion, and hence the momentum, of the electron. So a gamma-ray microscope would give you a highly accurate measurement of an electron's position, but the disturbance would create a less accurate measurement of the momentum. </p><p>
Heisenberg's thought experiment might seem clear but is misleading as it could suggest that the uncertainty principle is a result only of the disturbance caused by observation (this is another phenomenon, known as the <em><a href="https://en.wikipedia.org/wiki/Observer_effect_(physics)">observer effect</a></em>). "Heisenberg's uncertainty principle is often phrased in terms of measurement, but really you should think [in terms of] the electron itself," says <A href="http://www.damtp.cam.ac.uk/user/bca20/index.html">Ben Allanach</a>, professor of theoretical physics at the University of Cambridge. The uncertainty captured in the principle is inherent in the quantum world, whether it is observed or not.
</p>
<h3>A clearer view of uncertainty</h3>
<p>
Although Heisenberg's uncertainty principle might seem a confusing idea, it has a simple clarity when you describe the situation mathematically. A quantum system, such as the electron Heisenberg was thinking of, can be described mathematically using <a href="/content/ridiculously-brief-introduction-quantum-mechanics">Schrödinger's equation</a>. The state of that system at a particular time is given by the <em>wave function</em>, a solution to this equation, which can only give probabilities for the values of particular properties of the system. The fuzziness inherent in quantum physics means that it is impossible to predict the position of an electron when we measure it. "What you think of as an electron is fuzzy in space," says Allanach. "When you measure an electron you measure it to be in a precise position. But if you prepared a million identical electrons and measured them they'd be spread around a little bit." The spread of these measurements would reflect the probabilities given by the wavefunction. The same goes for any other thing you might want to measure about the particle, for example its momentum: all you can do is work out the probability that the momentum takes each of several possible values.
</p><p>
To work out from the wave function what those possible values of position and momentum are, you need mathematical objects called <em>operators</em>. There are many different operators, one for position, one for momentum, one for any property you are able to observe in a system. Usually the operator, such as the position operator, acts on a wave function to return a description of the possible locations of an electron and the probabilities you’d find the electron in each location if you were to measure it. But for each operator, there are particular wave functions called <em>eigenstates</em> of that operator, that return a single location, indicating that for that particular eigenstate the operator will return a single location at which the electron will be found with 100% certainty.
</p><p>
The same is true for other operators. There are eigenstates of the momentum operator, that will have a particular value for momentum, with 100% certainty. But it is a mathematical fact that a system can never be in an eigenstate of both the position and momentum operators simultaneously. Just as 3+2 will never make 27, so the mathematical operators corresponding to position and momentum don’t behave in a way that would allow them to have coinciding eigenstates. (For those familiar with some of the technicalities, the eigenstates cannot be the same because the operators don't commute.)
</p>
<div class="leftimage" style="max-width:300px">
<img src="/content/sites/plus.maths.org/files/articles/2018/measurement/jellly_web.jpg" alt="jelly"/><p>There's a limit to how much you can simultaneously squeeze the quantum fuzziness of an electron's position and momentum.</p></div>
<p>
The same goes for other complementary pairs of operators, such as those for time and energy, or for the quantum spin of a particle in different perpendicular directions. Mathematically, 100% certainty about the values of these pairs of properties just isn’t possible, regardless of measuring the quantum system. "That quantum fuzziness is like a physical limitation to the amount that you can specify where any electrons are going to be before you measure them," says Allanach. "It's really to do with the electron, with the particles themselves, rather than just measurement." You can't squeeze the quantum fuzziness of both the position and momentum properties of a particle below a certain limit simultaneously, regardless of if you measure these properties.
</p><p>
Since Heisenberg first described the uncertainty principle in terms of position and momentum, similar uncertainty principles have been described for other complementary pairs. All of these lead to fascinating and surprising results in quantum physics, such as the <a href="/content/problem-infinity">bubbling of virtual particles in a vacuum</a>. And don't worry if you still find these ideas counterintuitive, you are in very good company: <a href="/content/john-conway-discovering-free-will-part-i">Einstein himself</a> was appalled by Heisenberg's uncertainty principle and the "cloudy fitfulness" provided by this view of reality. Physicists today have gotten used to these ideas and are <a href="/content/play-quantum-lottery">making good use of this fuzziness</a>.
</p>
<hr><h3>About the article</h3>
<div class="leftimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2015/LHC/metalksmall.jpg" alt="" width="200" height="239" />
</div>
<p><a href="http://users.hepforge.org/~allanach/">Ben Allanach</a> is a Professor in Theoretical Physics at the <a href="http://www.damtp.cam.ac.uk/">Department of Theoretical Physics and Applied Mathematics</a> at the University of Cambridge. His research focuses on discriminating different models of particle physics using LHC data. He worked at CERN as a research fellow and continues to visit frequently.</p>
<p><a href="/content/people/index.html#rachel">Rachel Thomas</a> is Editor of <em>Plus</em>. </p>
<br class="brclear"/>
<div class="rightimage" style="max-width: 260px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/></div>
<p>This article is part of our <a href="/content/whos-watching-physics-observers"><em>Who's watching? The physics of observers</em> project</a>, run <a href="/content/whos-watching-physics-observers#fqxi">in collaboration with FQXi</a>. Click <a href="/content/whos-watching-limits-observation">here</a> to see more articles about the limits of observation.</p> </div></div></div>Wed, 27 Jun 2018 17:24:04 +0000Rachel7044 at https://plus.maths.org/contenthttps://plus.maths.org/content/heisenbergs-uncertainty-principle#comments