php form submitted, but no email received - php

Question

I am attempting to send an email by filling out a contact form. Both my html form and php file are included below. Once I fill out the form, it notifies me that the email has been sent. However, I never receive an email; I cannot figure out what i am doing wrong.
This is my html form:
<div class="row" style="padding-top:27px">
<div class="large-6 column">
<form action="mail.php" method="POST" data-abide>
<div class="name-field">
<label>Your name*</label>
<input type="text" required pattern="[a-zA-Z]+">
<small class="error">Name is required and must be a string.</small>
</div>
<div class="email-field">
<label>Email*</label>
<input type="email" required>
<small class="error">An email address is required.</small>
</div>
<div>
<label> Subject* </label>
<!--<input class="error" type="subject" name="subject" value="" size="40" required>
<small class="error">Invalid entry</small>-->
<select id="customDropdown1" class="medium error" required="" data-invalid="">
<option value="">Select an option...</option>
<option value="first">Green Chilies</option>
<option value="second">Raisins</option>
<option value="third">Panko bread crumbs</option>
<option value="fourth">Assistance</option>
</select>
<small class="error">Invalid entry</small>
</div>
</div>
<div class="large-6 column">
<!-- <label> Message </label>
<textarea class="error" placeholder="Message..." id="info" class="message" name="message" cols="70" rows="60" size="90" requi></textarea><br>-->
<label> Message </label>
<textarea class="error" placeholder="Message..." ></textarea>
<small class="error">Invalid entry</small>
<div>
<input style="float:right;" type="submit" value="Submit">
<img style="visibility: hidden;">
</div>
</form>
This is my php file:
<?php
$to = "education#arcdna.com";
$subject = $_POST['subject'];
$name = $_POST['name'];
$email = $_POST['email'];
$message = $_POST['message'];
$formcontent="From: $name \n Message: $message";
mail($to , $subject, $message, $email);
echo "Thank you for using our mail form";
?>
UPDATED FILES:
HTML:
<div class="name-field">
<label>Your name*</label>
<input type="text" name="name" required pattern="[a-zA-Z]+">
<small class="error">Name is required and must be a string.</small>
</div>
<div class="email-field">
<label>Email*</label>
<input type="email" name="email" required>
<small class="error">An email address is required.</small>
</div>
<div>
<label> Subject* </label>
<!--<input class="error" type="subject" name="subject" value="" size="40" required>
<small class="error">Invalid entry</small>-->
<select id="customDropdown1" class="medium error" name="subject" required="" data-invalid="">
<option value="">Select an option...</option>
<option value="first">Green Chilies</option>
<option value="second">Raisins</option>
<option value="third">Panko bread crumbs</option>
<option value="fourth">Assistance</option>
</select>
<small class="error">Invalid entry</small>
</div>
<!-- End Contact Form -->
</div>
PHP:
<?php
$to = "education#arcdna.com";
$subject = $_POST['subject'];
$name = $_POST['name'];
$email = $_POST['email'];
$message = $_POST['message'];
mail($to , $subject, $message, $email);
echo "Thank you for using our mail form";
?>

Answer 1

None of your input fields have a name associated with them, so you're likely getting empty data in your POST. The 'email' value within $_POST['email'] refers to an input field's name.
Specifically, you're not getting an email address from your form, so the email is not sending to anyone.
For instance, you'll need:
<input type="email" name="email" required>

Answer 2

The fourth parameter of mail() is additional headers. Which is where your "FROM" information should go.
Refer here:
http://php.net/manual/en/function.mail.php
for what should go there.
here's one example
$headers = 'From: webmaster#example.com' . "\r\n" .
'Reply-To: webmaster#example.com' . "\r\n" .
'X-Mailer: PHP/' . phpversion();