Your web-browser is very outdated, and as such, this website may not display properly. Please consider upgrading to a modern, faster and more secure browser. Click here to do so.

Mathmajik

Thinking Too Much
Apr 25 '14
Apr 25 '14
Monkeys Capable Of Mathematics
April 23, 2014 | by Stephen Luntz

photo credit: Margaret S. Livingstone. Rhesus monkeys can learn to associate symbols with quantity and that when it comes to rewards size matters

Next time you describe a job as so easy a trained monkey could do it, consider you might be underselling them. Rhesus monkeys have been found to be able to learn simple addition, and their rare errors may tell us something about how we estimate quantities ourselves.
Many animals have a sense of number. In his book The Mathematical Brain, Brian Butterworth describes experiments that have confirmed this across many species, such as one where lions were played the recorded roars of fellow Panthera leo. When the sounds came from fewer animals than made up the pride being tested they would set out to fight for their territory, but when the sounds suggested they were outnumbered they backed off.
Butterworth also has a story about a troupe of chimpanzees that communicated over large distances by hitting hollow trees, with the leader sending a messages in code – one strike meaning change direction, two for rest.
The capacity to add and subtract numbers together in symbolic form is a different matter however, although some past studies have suggested some primates may be capable of this too.
Professor Margaret Livingstone of the Harvard Medical School taught rhesus monkeys the meaning of numbers from zero to 25 using the symbols for 0-9 and 16 letters of the alphabet. The teaching was done using drops of a reward so that the larger number was associated with more drops. The monkeys were then given a choice of two symbols and given a number of drops of the reward equal to the side they chose. Consequently, it was in their interest to choose the larger side.
Once this had been learned successfully the monkeys were given two symbols and had to compare them with a single one. At first the monkeys were inclined to choose the side with the single number, if it was larger than either numbers on the other side on their own. With time they got better, realizing that two smaller numbers could combined be better than one larger one. Livingstone reports in the Proceedings of the National Academy of Sciences that the animals were successful 90% of the time. Interestingly, however, the monkeys still placed more weight on the larger number than the smaller one – that is they were more likely to pick a side with a 2 and a 9, when compared with a 10, than a 4 and 7, even though both add to 11.
Past tests of animal intelligence have often run into trouble because certain animals reached the same conclusion through a different method from what was expected. The team worried that through prolonged practice the monkeys might have memorized all possible pairings, demonstrating an extraordinary memory rather than any capacity for calculation. So Livingstone and her colleagues gave their subjects a new set of characters and taught them what each meant. Without further prompting the monkeys started using maths to work out which combinations were larger.
A further intriguing insight into the monkey mind came from the observations that when the monkeys did get it wrong it was usually when the totals were close together; 6+7 was hard when compared to 12, but easy when compared to 9. Rather than having a precise calculation the mathemagicians were using estimates, which may offer insight into the way humans do the same calculations.
Livingstone is hoping to find the basis for Weber’s Law, which says that how large the difference between two stimuli has to be for us to notice it depends not on the absolute size of the difference, but on the size relative to the magnitude of the stimuli. As the paper puts it, “Although it is easy to recognize the difference between 2 and 4 items, it is more difficult to distinguish 22 and 24 items.” The same occurs when we are trying to distinguish the size or weight of an object, or in estimating periods of time.
Livingstone and her co-authors say, “Weber’s law can be explained either by a compressive scaling of sensory response with stimulus magnitude or by a proportional scaling of response variability. These two mechanisms can be distinguished by asking how quantities are added or subtracted.” The authors conclude,  “The way [the monkeys] combined pairs of symbols indicated neither a linear nor a compressed scale, but rather a dynamically shifting, relative scaling.”

Read more at http://www.iflscience.com/brain/monkeys-capable-mathematics#yILTswR15hPqS3ML.99

Monkeys Capable Of Mathematics
April 23, 2014 | by Stephen Luntz

photo credit: Margaret S. Livingstone. Rhesus monkeys can learn to associate symbols with quantity and that when it comes to rewards size matters

Next time you describe a job as so easy a trained monkey could do it, consider you might be underselling them. Rhesus monkeys have been found to be able to learn simple addition, and their rare errors may tell us something about how we estimate quantities ourselves.

Many animals have a sense of number. In his book The Mathematical Brain, Brian Butterworth describes experiments that have confirmed this across many species, such as one where lions were played the recorded roars of fellow Panthera leo. When the sounds came from fewer animals than made up the pride being tested they would set out to fight for their territory, but when the sounds suggested they were outnumbered they backed off.

Butterworth also has a story about a troupe of chimpanzees that communicated over large distances by hitting hollow trees, with the leader sending a messages in code – one strike meaning change direction, two for rest.

The capacity to add and subtract numbers together in symbolic form is a different matter however, although some past studies have suggested some primates may be capable of this too.

Professor Margaret Livingstone of the Harvard Medical School taught rhesus monkeys the meaning of numbers from zero to 25 using the symbols for 0-9 and 16 letters of the alphabet. The teaching was done using drops of a reward so that the larger number was associated with more drops. The monkeys were then given a choice of two symbols and given a number of drops of the reward equal to the side they chose. Consequently, it was in their interest to choose the larger side.

Once this had been learned successfully the monkeys were given two symbols and had to compare them with a single one. At first the monkeys were inclined to choose the side with the single number, if it was larger than either numbers on the other side on their own. With time they got better, realizing that two smaller numbers could combined be better than one larger one. Livingstone reports in the Proceedings of the National Academy of Sciences that the animals were successful 90% of the time. Interestingly, however, the monkeys still placed more weight on the larger number than the smaller one – that is they were more likely to pick a side with a 2 and a 9, when compared with a 10, than a 4 and 7, even though both add to 11.

Past tests of animal intelligence have often run into trouble because certain animals reached the same conclusion through a different method from what was expected. The team worried that through prolonged practice the monkeys might have memorized all possible pairings, demonstrating an extraordinary memory rather than any capacity for calculation. So Livingstone and her colleagues gave their subjects a new set of characters and taught them what each meant. Without further prompting the monkeys started using maths to work out which combinations were larger.

A further intriguing insight into the monkey mind came from the observations that when the monkeys did get it wrong it was usually when the totals were close together; 6+7 was hard when compared to 12, but easy when compared to 9. Rather than having a precise calculation the mathemagicians were using estimates, which may offer insight into the way humans do the same calculations.

Livingstone is hoping to find the basis for Weber’s Law, which says that how large the difference between two stimuli has to be for us to notice it depends not on the absolute size of the difference, but on the size relative to the magnitude of the stimuli. As the paper puts it, “Although it is easy to recognize the difference between 2 and 4 items, it is more difficult to distinguish 22 and 24 items.” The same occurs when we are trying to distinguish the size or weight of an object, or in estimating periods of time.

Livingstone and her co-authors say, “Weber’s law can be explained either by a compressive scaling of sensory response with stimulus magnitude or by a proportional scaling of response variability. These two mechanisms can be distinguished by asking how quantities are added or subtracted.” The authors conclude, “The way [the monkeys] combined pairs of symbols indicated neither a linear nor a compressed scale, but rather a dynamically shifting, relative scaling.”

Read more at http://www.iflscience.com/brain/monkeys-capable-mathematics#yILTswR15hPqS3ML.99

Apr 24 '14
postminimaldubtech:

Fuck yeah calculus

postminimaldubtech:

Fuck yeah calculus

Apr 24 '14
"plus C"
ancient calculus proverb (via cosmicbeachparty)
Apr 24 '14
klimkovsky:

Graphs of different types of functions!

klimkovsky:

Graphs of different types of functions!

Apr 20 '14

third-eyes:

staceythinx:

Science-inspired necklaces from the Delftia Etsy store

Apr 20 '14
COMPUTUS:

Computus (Latin for “computation”) is the calculation of the date of Easter in terms of, first, the Julian and, later, the Gregorian calendar. The name has been used for this procedure since the early Middle Ages, as it was considered the most important computation of the age.

Following the First Council of Nicaea, the date for Easter was completely divorced from the Jewish calendar and its computations for Passover. Thereafter, in principle, Easter fell on the Sunday following the full moon that follows the Northern spring equinox (the so-called Paschal Full Moon). However, the vernal equinox and the full moon were not determined by astronomical observation. Instead, the vernal equinox was fixed to fall on the 21st day of March, while the full moon (known as the ecclesiastical full moon) was fixed at 14 days after the beginning of the ecclesiastical lunar month (known as the ecclesiastical new moon). Easter thus falls on the Sunday after the ecclesiastical full moon. The computus is the procedure of determining the first Sunday after the first ecclesiastical full moon falling on or after 21 March and the difficulty arose from doing this over the span of centuries without accurate means of measuring the precise solar or lunar years.

The model that was worked out assumes that 19 tropical years have the same duration as 235 synodic months (modern value: 234.997).[1]

Since the 16th century, there have been differences in the calculation of Easter between the Western and Eastern Churches. The Roman Catholic Church since 1583 has been using 21 March under the Gregorian calendar to calculate the date of Easter, while the Eastern Orthodox continues to use 21 March under the Julian Calendar. The Catholic and Protestant denominations thus use an ecclesiastical full moon that occurs four to five days earlier than the eastern one.

peltiertech.com

COMPUTUS:

Computus (Latin for “computation”) is the calculation of the date of Easter in terms of, first, the Julian and, later, the Gregorian calendar. The name has been used for this procedure since the early Middle Ages, as it was considered the most important computation of the age.

Following the First Council of Nicaea, the date for Easter was completely divorced from the Jewish calendar and its computations for Passover. Thereafter, in principle, Easter fell on the Sunday following the full moon that follows the Northern spring equinox (the so-called Paschal Full Moon). However, the vernal equinox and the full moon were not determined by astronomical observation. Instead, the vernal equinox was fixed to fall on the 21st day of March, while the full moon (known as the ecclesiastical full moon) was fixed at 14 days after the beginning of the ecclesiastical lunar month (known as the ecclesiastical new moon). Easter thus falls on the Sunday after the ecclesiastical full moon. The computus is the procedure of determining the first Sunday after the first ecclesiastical full moon falling on or after 21 March and the difficulty arose from doing this over the span of centuries without accurate means of measuring the precise solar or lunar years.

The model that was worked out assumes that 19 tropical years have the same duration as 235 synodic months (modern value: 234.997).[1]

Since the 16th century, there have been differences in the calculation of Easter between the Western and Eastern Churches. The Roman Catholic Church since 1583 has been using 21 March under the Gregorian calendar to calculate the date of Easter, while the Eastern Orthodox continues to use 21 March under the Julian Calendar. The Catholic and Protestant denominations thus use an ecclesiastical full moon that occurs four to five days earlier than the eastern one.

peltiertech.com
Apr 20 '14
cheak:

givemeinternet:

The closer to the end the more satisfying it gets…


omg i have this sunglasses

cheak:

givemeinternet:

The closer to the end the more satisfying it gets…

omg i have this sunglasses

Apr 19 '14
centerofmath:

The Ulam spiral shows a pattern in the (unpredictable) primes

centerofmath:

The Ulam spiral shows a pattern in the (unpredictable) primes

Apr 18 '14
hipster-graphs:

1: sinx=cosy
2: 2siny=cosx
3: sinx=cosy+1
4: sinx=cosy+2
5: sinx=cosy+1

hipster-graphs:

1: sinx=cosy

2: 2siny=cosx

3: sinx=cosy+1

4: sinx=cosy+2

5: sinx=cosy+1

Apr 17 '14

jtotheizzoe:

spacetravelco:

Scientific engravings from 1850

by John Philipps Emslie

(via the Wellcome Collection)

I move to give John Philipps Emslie his own posthumous Tumblr, like now.

Apr 17 '14
superloglady:

Jean François Niceron | La Perspective Curieuse
@BibliOdyssey

superloglady:

Jean François Niceron | La Perspective Curieuse

@BibliOdyssey

Apr 17 '14
Apr 14 '14

(Source: studygeek)

Map